In Order for the Figure to Map Onto Itself, the Line of Reflection Must Go Through the Center Point

Home , Rotational symmetry

State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

Thus, there are four possible lines that go through the center and are lines of reflections.

3-5 Symmetry Therefore, the figure has four lines of symmetry.

State whether the figure appears to have line ANSWER: symmetry. Write yes or no. If so, copy the yes; 4 figure, draw all lines of symmetry, and state their number.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. 2.

In order for the figure to map onto itself, the line of SOLUTION: reflection must go through the center point. A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line.

Two lines of reflection go through the sides of the The given figure does not have reflectional figure. symmetry. There is no way to fold or reflect it onto itself.

ANSWER: no

Two lines of reflection go through the vertices of the figure. 3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

The figure has a vertical line of symmetry. Thus, there are four possible lines that go through the center and are lines of reflections.

It does not have a horizontal line of symmetry.

Therefore, the figure has four lines of symmetry. eSolutionsANSWER: Manual - Powered by Cognero Page 1 yes; 4 The figure does not have a line of symmetry through the vertices.

Thus, the figure has only one line of symmetry.

ANSWER: 2. yes; 1 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself. State whether the figure has rotational symmetry. Write yes or no. If so, copy the ANSWER: figure, locate the center of symmetry, and state no the order and magnitude of symmetry.

The given figure has reflectional symmetry. For the given figure, there is no rotation between 0°

and 360° that maps the figure onto itself. If the figure The figure has a vertical line of symmetry. were a regular pentagon, it would have rotational symmetry.

ANSWER: no

It does not have a horizontal line of symmetry.

5. SOLUTION:

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The figure does not have a line of symmetry through between 0° and 360° about the center of the figure. the vertices. The given figure has rotational symmetry.

Thus, the figure has only one line of symmetry.

ANSWER:

yes; 1 The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of symmetry.

The given figure has order of symmetry of 2, since the figure can be rotated twice in 360°.

State whether the figure has rotational The magnitude of symmetry is the smallest angle symmetry. Write yes or no. If so, copy the through which a figure can be rotated so that it maps figure, locate the center of symmetry, and state onto itself. the order and magnitude of symmetry. Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is .

ANSWER: 4. yes; 2; 180° SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: 6. no SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry. 5. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps

onto itself. The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of The figure has magnitude of symmetry of symmetry. .

The given figure has order of symmetry of 2, since ANSWER: the figure can be rotated twice in 360°. yes; 4; 90°

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is . State whether the figure has line symmetry ANSWER: and/or rotational symmetry. If so, describe the yes; 2; 180° reflections and/or rotations that map the figure onto itself.

7. SOLUTION: 6. Vertical and horizontal lines through the center and SOLUTION: diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The equations of those lines in this figure are x = 0,

between 0° and 360° about the center of the figure. y = -1, y = x - 1, and y = -x - 1.

The given figure has rotational symmetry. Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in The number of times a figure maps onto itself as it the line y = -x - 1 map the square onto itself; the rotates from 0° to 360° is called the order of rotations of 90, 180, and 270 degrees around the point

symmetry. (0, -1) map the square onto itself.

Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. 8. The figure has magnitude of symmetry of . SOLUTION: This figure does not have line symmetry, because ANSWER: adjacent sides are not congruent. yes; 4; 90° It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself. State whether the figure has line symmetry and/or rotational symmetry. If so, describe the REGULARITY State whether the figure reflections and/or rotations that map the figure appears to have line symmetry. Write yes or no. onto itself. If so, copy the figure, draw all lines of symmetry, and state their number.

9. 7. SOLUTION: A figure has reflectional symmetry if the figure can SOLUTION: be mapped onto itself by a reflection in a line. Vertical and horizontal lines through the center and diagonal lines through two opposite vertices are all For the given figure, there are no lines of reflection lines of symmetry for a square oriented this way. where the figure can map onto itself. Thus, the figure

does not have any lines of of symmetry. The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1. ANSWER: no Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the 10. reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the SOLUTION: rotations of 90, 180, and 270 degrees around the point A figure has reflectional line symmetry if the figure (0, -1) map the square onto itself. can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

The figure has a vertical and horizontal line of reflection. 8. SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself. It is also possible to have reflection over the diagonal lines. ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

REGULARITY State whether the figure appears to have line symmetry. Write yes or no. Therefore, the figure has four lines of symmetry If so, copy the figure, draw all lines of symmetry, and state their number.

9. SOLUTION:

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. ANSWER: yes; 4 For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry.

ANSWER: no

10.

SOLUTION: 11. A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line. SOLUTION: A figure has reflectional symmetry if the figure can The given figure has reflectional symmetry. be mapped onto itself by a reflection in a line.

In order for the figure to map onto itself, the line of The given hexagon has reflectional symmetry. reflection must go through the center point. In order for the hexagon to map onto itself, the line The figure has a vertical and horizontal line of of reflection must go through the center point. reflection. There are three lines of reflection that go though opposites edges.

It is also possible to have reflection over the diagonal lines. There are three lines of reflection that go though opposites vertices.

Therefore, the figure has four lines of symmetry

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry.

ANSWER: yes; 4

ANSWER: yes; 6

11. SOLUTION: A figure has reflectional symmetry if the figure can 12. be mapped onto itself by a reflection in a line. SOLUTION: The given hexagon has reflectional symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. In order for the hexagon to map onto itself, the line of reflection must go through the center point. The figure has reflectional symmetry.

There are three lines of reflection that go though There is only one line of symmetry, a horizontal line opposites edges. through the middle of the figure.

Thus, the figure has one line of symmetry.

ANSWER: There are three lines of reflection that go though yes; 1 opposites vertices.

13. There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has SOLUTION: six lines of symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

There is only one possible line of reflection, horizontally though the middle of the figure.

ANSWER: yes; 6 Thus, the figure has one line of symmetry. ANSWER: yes; 1

14. 12. SOLUTION: SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The given figure does not have reflectional symmetry. It is not possible to draw a line of The figure has reflectional symmetry. reflection where the figure can map onto itself.

There is only one line of symmetry, a horizontal line ANSWER: through the middle of the figure. no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their number. Thus, the figure has one line of symmetry. 15. Refer to page 262. ANSWER: SOLUTION: yes; 1 A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

13. ANSWER: no SOLUTION: A figure has reflectional symmetry if the figure can 16. Refer to the flag on page 262. be mapped onto itself by a reflection in a line. SOLUTION: The figure has reflectional symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. There is only one possible line of reflection, horizontally though the middle of the figure. The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Thus, the figure has one line of symmetry. A horizontal and vertical lines of reflection are ANSWER: possible. yes; 1

14.

SOLUTION: Two diagonal lines of reflection are possible. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. It is not possible to draw a line of reflection where the figure can map onto itself.

ANSWER: no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the There are a total of four possible lines that go flag, draw all lines of symmetry, and state their through the center and are lines of reflections. Thus, number. the flag has four lines of symmetry. 15. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry. ANSWER: ANSWER: no yes; 4

16. Refer to the flag on page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point. 17. Refer to page 262. A horizontal and vertical lines of reflection are possible. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

Two diagonal lines of reflection are possible.

It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

There are a total of four possible lines that go through the center and are lines of reflections. Thus,

the flag has four lines of symmetry. State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

18. SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the yes; 4 figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of 17. Refer to page 262. symmetry.

SOLUTION: This figure has order 2 rotational symmetry, since A figure has reflectional symmetry if the figure can you have to rotate 180° to get the figure to map onto be mapped onto itself by a reflection in a line. itself.

The figure has reflectional symmetry. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps A horizontal line is a line of reflections for this flag. onto itself.

The figure has a magnitude of symmetry of .

ANSWER:

It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry. yes; 2; 180° ANSWER: yes; 1

19. SOLUTION: A figure in the plane has rotational symmetry if the State whether the figure has rotational figure can be mapped onto itself by a rotation symmetry. Write yes or no. If so, copy the between 0° and 360° about the center of the figure. figure, locate the center of symmetry, and state the order and magnitude of symmetry. The triangle has rotational symmetry.

18. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle The figure has rotational symmetry. through which a figure can be rotated so that it maps onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The figure has magnitude of symmetry of symmetry. .

This figure has order 2 rotational symmetry, since ANSWER: you have to rotate 180° to get the figure to map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. yes; 3; 120°

The figure has a magnitude of symmetry of .

ANSWER: 20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. yes; 2; 180° The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

19. ANSWER: SOLUTION: no A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The triangle has rotational symmetry. 21.

SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The crescent shaped figure has no rotational

symmetry. There is no way to rotate it such that it can be mapped onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 3; 120°

20. SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the no figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be 22. mapped onto itself. SOLUTION: A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation no between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation

between 0° and 360° about the center of the figure.

The number of times a figure maps onto itself as it The crescent shaped figure has no rotational rotates from 0° to 360° is called the order of symmetry. There is no way to rotate it such that it symmetry.

can be mapped onto itself. The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 8; 45°

23. ANSWER: SOLUTION: no A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry. 22. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This means that the figure can be rotated 8 times and map onto itself within 360°.

symmetry. The figure has magnitude of symmetry of . The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it ANSWER: map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. yes; 8; 45° The figure has magnitude of symmetry of . WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, ANSWER: state the order and magnitude of symmetry. 24. Refer to page 263. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

yes; 8; 45° The wheel has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. 23. The wheel has order 5 rotational symmetry. There SOLUTION: are 5 large spokes and 5 small spokes. You can A figure in the plane has rotational symmetry if the rotate the wheel 5 times within 360° and map the figure can be mapped onto itself by a rotation figure onto itself. between 0° and 360° about the center of the figure. The magnitude of symmetry is the smallest angle The figure has rotational symmetry. through which a figure can be rotated so that it maps onto itself.

The wheel has magnitude of symmetry .

ANSWER:

The number of times a figure maps onto itself as it yes; 5; 72° rotates from 0° to 360° is called the order of symmetry. 25. Refer to page 263.

The figure has order 8 rotational symmetry. This SOLUTION: means that the figure can be rotated 8 times and map A figure in the plane has rotational symmetry if the onto itself within 360°. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The wheel has rotational symmetry. onto itself. The number of times a figure maps onto itself as it The figure has magnitude of symmetry of rotates from 0° to 360° is called the order of . symmetry.

ANSWER: The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps yes; 8; 45° onto itself.

WHEELS State whether each wheel cover appears The wheel has order 8 rotational symmetry and to have rotational symmetry. Write yes or no. If so, . state the order and magnitude of symmetry. magnitude 24. Refer to page 263. ANSWER: SOLUTION: yes; 8; 45° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 26. Refer to page 263. between 0° and 360° about the center of the figure. SOLUTION: A figure in the plane has rotational symmetry if the The wheel has rotational symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The number of times a figure maps onto itself as it The wheel has rotational symmetry. rotates from 0° to 360° is called the order of symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The wheel has order 5 rotational symmetry. There symmetry. The wheel has order 10 rotational are 5 large spokes and 5 small spokes. You can symmetry. There are 10 bolts and the tire can be rotate the wheel 5 times within 360° and map the rotated 10 times within 360° and map onto itself. figure onto itself. The magnitude of symmetry is the smallest angle The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps through which a figure can be rotated so that it maps onto itself. The wheel has magnitude of symmetry of onto itself. .

The wheel has magnitude of symmetry ANSWER: . yes; 10; 36° ANSWER: State whether the figure has line symmetry yes; 5; 72° and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself. 25. Refer to page 263. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The wheel has rotational symmetry. 27. The number of times a figure maps onto itself as it SOLUTION: rotates from 0° to 360° is called the order of This triangle is scalene, so it cannot have symmetry. symmetry. ANSWER: The wheel has order 8 rotational symmetry. There no symmetry are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The wheel has order 8 rotational symmetry and 28. magnitude . SOLUTION: ANSWER: This figure is a square, because each pair of adjacent yes; 8; 45° sides is congruent and perpendicular. All squares have both line and rotational symmetry. 26. Refer to page 263. The line symmetry is vertically, horizontally, and SOLUTION: diagonally through the center of the square, with lines that are either parallel to the sides of the square or A figure in the plane has rotational symmetry if the that include two vertices of the square. The figure can be mapped onto itself by a rotation equations of those lines are: x = 0, y = 0, y = x, and y between 0° and 360° about the center of the figure. = -x The wheel has rotational symmetry.

The rotational symmetry is for each quarter turn in a The number of times a figure maps onto itself as it square, so the rotations of 90, 180, and 270 degrees rotates from 0° to 360° is called the order of around the origin map the square onto itself. symmetry. The wheel has order 10 rotational symmetry. There are 10 bolts and the tire can be rotated 10 times within 360° and map onto itself. ANSWER: line symmetry; rotational symmetry; the reflection in The magnitude of symmetry is the smallest angle the line x = 0, the reflection in the line y = 0, the through which a figure can be rotated so that it maps reflection in the line y = x, and the reflection in the onto itself. The wheel has magnitude of symmetry of line y = -x all map the square onto itself; the rotations . of 90, 180, and 270 degrees around the origin map the square onto itself. ANSWER: yes; 10; 36° State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself. 29. SOLUTION: The trapezoid has line symmetry, because it is isosceles, but it does not have rotational symmetry, because no trapezoid does.

27. The reflection in the line y = 1.5 maps the trapezoid onto itself, because that is the perpendicular bisector SOLUTION: to the parallel sides. This triangle is scalene, so it cannot have symmetry. ANSWER: ANSWER: line symmetry; the reflection in the line y = 1.5 maps no symmetry the trapezoid onto itself.

28. 30. SOLUTION: SOLUTION: This figure is a square, because each pair of adjacent This figure is a parallelogram, so it has rotational sides is congruent and perpendicular. symmetry of a half turn or 180 degrees around its All squares have both line and rotational symmetry. center, which is the point (1, -1.5). The line symmetry is vertically, horizontally, and diagonally through the center of the square, with lines Since this parallelogram is not a rhombus it does not that are either parallel to the sides of the square or have line symmetry. that include two vertices of the square. The equations of those lines are: x = 0, y = 0, y = x, and y ANSWER: = -x rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram The rotational symmetry is for each quarter turn in a onto itself. square, so the rotations of 90, 180, and 270 degrees around the origin map the square onto itself. 31. MODELING Symmetry is an important component of photography. Photographers often use reflection in water to create symmetry in photos. The photo on ANSWER: page 263 is a long exposure shot of the Eiffel tower line symmetry; rotational symmetry; the reflection in reflected in a pool. the line x = 0, the reflection in the line y = 0, the reflection in the line y = x, and the reflection in the a. Describe the two-dimensional symmetry created line y = -x all map the square onto itself; the rotations by the photo. of 90, 180, and 270 degrees around the origin map b. Is there rotational symmetry in the photo? Explain the square onto itself. your reasoning. SOLUTION: a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is

29. reflected over the horizontal line, there is no SOLUTION: rotational symmetry. The trapezoid has line symmetry, because it is ANSWER: isosceles, but it does not have rotational symmetry, because no trapezoid does. a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. The reflection in the line y = 1.5 maps the trapezoid There is a vertical line of symmetry through the onto itself, because that is the perpendicular bisector center of the photo. to the parallel sides. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no ANSWER: rotational symmetry. line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself. COORDINATE GEOMETRY Determine whether the figure with the given vertices has line symmetry and/or rotational symmetry. 32. R(–3, 3), S(–3, –3), T(3, 3) SOLUTION: Draw the figure on a coordinate plane.

30. SOLUTION: This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its center, which is the point (1, -1.5).

Since this parallelogram is not a rhombus it does not have line symmetry. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. ANSWER: rotational symmetry; the rotation of 180 degrees The given triangle has a line of symmetry through around the point (1, -1.5) maps the parallelogram points (0, 0) and (–3, 3). onto itself. A figure in the plane has rotational symmetry if the 31. MODELING Symmetry is an important component figure can be mapped onto itself by a rotation of photography. Photographers often use reflection in between 0° and 360° about the center of the figure. water to create symmetry in photos. The photo on There is not way to rotate the figure and have it map page 263 is a long exposure shot of the Eiffel tower onto itself. reflected in a pool.

Thus, the figure has only line symmetry. a. Describe the two-dimensional symmetry created by the photo. ANSWER: b. Is there rotational symmetry in the photo? Explain line your reasoning. SOLUTION: 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) a Sample answer: There is a horizontal line of SOLUTION: symmetry between the tower and its reflection. There is a vertical line of symmetry through the Draw the figure on a coordinate plane. center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no rotational symmetry.

ANSWER: a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. A figure has line symmetry if the figure can be b No; sample answer: Because of how the image is mapped onto itself by a reflection in a line. The reflected over the horizontal line, there is no given figure has 4 lines of symmetry. The line of rotational symmetry. symmetry are though the following pairs of points {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, COORDINATE GEOMETRY Determine and {(2, 2), (2, –2)}. whether the figure with the given vertices has line symmetry and/or rotational symmetry. A figure in the plane has rotational symmetry if the 32. R(–3, 3), S(–3, –3), T(3, 3) figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. SOLUTION: The figure can be rotated from the origin and map Draw the figure on a coordinate plane. onto itself. The order of symmetry is 4.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational

34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 2) A figure has line symmetry if the figure can be SOLUTION: mapped onto itself by a reflection in a line. Draw the figure on a coordinate plane. The given triangle has a line of symmetry through points (0, 0) and (–3, 3).

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. There is not way to rotate the figure and have it map onto itself. A figure has line symmetry if the figure can be Thus, the figure has only line symmetry. mapped onto itself by a reflection in a line. The given hexagon has 2 lines of symmetry. The lines ANSWER: pass through the following pair of points {(0, 4), (0, – line 4)}, and {(3, 0), (–3, 0)}

33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. Draw the figure on a coordinate plane. The figure has rotational symmetry. You can rotate the figure once within 360° and have it map to itself.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3)

A figure has line symmetry if the figure can be SOLUTION: mapped onto itself by a reflection in a line. The Draw the figure on a coordinate plane. given figure has 4 lines of symmetry. The line of symmetry are though the following pairs of points {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, and {(2, 2), (2, –2)}.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The Thus, the figure has both line symmetry and trapezoid has a line of reflection through points (0,3) rotational symmetry. and (0, –3). ANSWER: line and rotational A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – between 0° and 360° about the center of the 2) figure. There is no way to rotate this figure and have it map onto itself. Thus, it does not have rotational SOLUTION: symmetry. Draw the figure on a coordinate plane. Therefore, the figure has only line symmetry.

ANSWER: line

ALGEBRA Graph the function and determine whether the graph has line and/or rotational symmetry. If so, state the order and magnitude of symmetry, and write the equations of any lines of A figure has line symmetry if the figure can be symmetry. mapped onto itself by a reflection in a line. The 36. y = x given hexagon has 2 lines of symmetry. The lines pass through the following pair of points {(0, 4), (0, – SOLUTION: 4)}, and {(3, 0), (–3, 0)} Graph the function.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: A figure has reflectional symmetry if the figure can line and rotational be mapped onto itself by a reflection in a line. The line y = x has reflectional symmetry since any line 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) perpendicular to y = x is a line of reflection. The SOLUTION: equation of the line symmetry is y = –x. Draw the figure on a coordinate plane. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The line can be rotated twice within 360° and be mapped onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of

symmetry. The graph has order 2 rotational symmetry. A figure has line symmetry if the figure can be

mapped onto itself by a reflection in a line. The trapezoid has a line of reflection through points (0,3) The magnitude of symmetry is the smallest angle and (0, –3). through which a figure can be rotated so that it maps

onto itself. The graph has magnitude of symmetry of A figure in the plane has rotational symmetry if the . figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. There is no way to rotate this figure and have Thus, the graph has both reflectional and rotational it map onto itself. Thus, it does not have rotational symmetry. symmetry. ANSWER: Therefore, the figure has only line symmetry. rotational; 2; 180°; line symmetry; y = –x

ANSWER: line

ALGEBRA Graph the function and determine whether the graph has line and/or rotational symmetry. If so, state the order and magnitude of symmetry, and write the equations of any lines of symmetry.

36. y = x

2 SOLUTION: 37. y = x + 1 Graph the function. SOLUTION: Graph the function.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation A figure in the plane has rotational symmetry if the between 0° and 360° about the center of the figure. figure can be mapped onto itself by a rotation The line can be rotated twice within 360° and be between 0° and 360° about the center of the mapped onto itself. figure. There is no way to rotate the graph and have

it map onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of Thus, the graph has only reflectional symmetry. symmetry. The graph has order 2 rotational ANSWER: symmetry. line; x = 0 The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of .

Thus, the graph has both reflectional and rotational symmetry.

ANSWER: 38. y = –x3 rotational; 2; 180°; line symmetry; y = –x SOLUTION: Graph the function.

2 37. y = x + 1

SOLUTION: A figure has reflectional symmetry if the figure can Graph the function. be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself.

The number of times a figure maps onto itself as it A figure has reflectional symmetry if the figure can rotates from 0° to 360° is called the order of be mapped onto itself by a reflection in a line. The symmetry. The graph has order 2 rotational graph is reflected through the y-axis. Thus, the symmetry. equation of the line symmetry is x = 0. The magnitude of symmetry is the smallest angle A figure in the plane has rotational symmetry if the through which a figure can be rotated so that it maps figure can be mapped onto itself by a rotation onto itself. The graph has magnitude of symmetry of between 0° and 360° about the center of the . figure. There is no way to rotate the graph and have it map onto itself. Thus, the graph has only rotational symmetry.

Thus, the graph has only reflectional symmetry. ANSWER: rotational; 2; 180° ANSWER: line; x = 0

39. Refer to the rectangle on the coordinate plane.

38. y = –x3 SOLUTION: Graph the function.

A figure has reflectional symmetry if the figure can SOLUTION: be mapped onto itself by a reflection in a line. The a. The lines of symmetry are parallel to the sides of graph does not have a line of reflections where the the rectangles, and through the center of rotation.

graph can be mapped onto itself.

The slopes of the sides of the rectangle are 0.5 and A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation -2, so the slopes of the lines of symmetry are the between 0° and 360° about the center of the same. figure. You can rotate the graph through the origin The center of the rectangle is (1, 1.5). Use the and have it map onto itself. point-slope formula to find equations.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of

symmetry. The graph has order 2 rotational b. The equations of the lines of symmetry do not symmetry. change; although the rectangle does not map onto

The magnitude of symmetry is the smallest angle itself under this rotation, the lines of symmetry are through which a figure can be rotated so that it maps mapped to each other. The graph has magnitude onto itself. of symmetry of ANSWER: . a.

b. The equations of the lines of symmetry do not Thus, the graph has only rotational symmetry. change; although the rectangle does not map onto ANSWER: itself under this rotation, the lines of symmetry are rotational; 2; 180° mapped to each other. 40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

d. Verbal Make a conjecture about the number of a. What are the equations of the lines of symmetry of lines of symmetry and the order of symmetry for a

the rectangle? regular polygon with n sides. b. What happens to the equations of the lines of symmetry when the rectangle is rotated 90 degrees SOLUTION: counterclockwise around its center of symmetry? a. Construct an equilateral triangle and label the Explain. vertices A, B, and C. Draw a line through A perpendicular to . Reflect the triangle in the line. SOLUTION: Show the labels of the reflected image. If the image a. The lines of symmetry are parallel to the sides of maps to the original, then this line is a line of the rectangles, and through the center of rotation. reflection.

The slopes of the sides of the rectangle are 0.5 and -2, so the slopes of the lines of symmetry are the same. The center of the rectangle is (1, 1.5). Use the point-slope formula to find equations.

b. The equations of the lines of symmetry do not Next, draw a line through B perpendicular to . change; although the rectangle does not map onto Reflect the triangle in the line. Show the labels of the itself under this rotation, the lines of symmetry are reflected image. If the image maps to the original, mapped to each other. then this line is a line of reflection.

ANSWER: a. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to Lastly, draw a line through C perpendicular to . investigate line and rotational symmetry in regular Reflect the triangle in the line. Show the labels of the polygons. reflected image. If the image maps to the original, then this line is a line of reflection. a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides. SOLUTION: a. Construct an equilateral triangle and label the There are 3 lines of symmetry. vertices A, B, and C. Draw a line through A b. Construct an equilateral triangle and show the perpendicular to . Reflect the triangle in the line. labels of the vertices. Next, find the center of the Show the labels of the reflected image. If the image triangle. Since this is an equilateral triangle, the maps to the original, then this line is a line of circumcenter, incenter, centroid, and orthocenter are reflection. the same point. Construct altitudes through each vertex and label the intersection. Rotate the triangle about point D. A 120 degree rotation will map the image to the original. Show the labels of the image.

Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

There are 3 lines of symmetry.

b. Construct an equilateral triangle and show the labels of the vertices. Next, find the center of the

triangle. Since this is an equilateral triangle, the Since the figure maps onto itself 3 times as it is circumcenter, incenter, centroid, and orthocenter are rotated, the order of symmetry is 3. the same point. Construct altitudes through each c. vertex and label the intersection. Square Rotate the triangle about point D. A 120 degree Construct a square and then construct lines through rotation will map the image to the original. Show the the midpoints of each side and diagonals. Use the labels of the image. reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5. The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

Since the figure maps onto itself 3 times as it is Regular Hexagon rotated, the order of symmetry is 3. Construct a regular hexagon and then construct lines c. through each vertex perpendicular to the sides. Use Square the reflection tool first to find that the image maps Construct a square and then construct lines through onto the original when reflected in each of the 6 lines the midpoints of each side and diagonals. Use the constructed. So there are 6 lines of symmetry. reflection tool first to find that the image maps onto Next, rotate the square about the center point. The the original when reflected in each of the 4 lines image maps to the original at 60, 120, 180, 240, 300, constructed. So there are 4 lines of symmetry. and 360 degree rotations. So the order of symmetry Next, rotate the square about the center point. The is 6. image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Regular Pentagon d. Sample answer: for each figure studied, the Construct a regular pentagon and then construct lines number of sides of the figure is the same as the lines through each vertex perpendicular to the sides. Use of symmetry and the order of symmetry. A regular the reflection tool first to find that the image maps polygon with n sides has n lines of symmetry and onto the original when reflected in each of the 5 lines order of symmetry n. constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The ANSWER: image maps to the original at 72, 144, 216, 288, and a. 3 360 degree rotations. So the order of symmetry is 5. b. 3

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

Regular Hexagon Construct a regular hexagon and then construct lines through each vertex perpendicular to the sides. Use SOLUTION: the reflection tool first to find that the image maps onto the original when reflected in each of the 6 lines A figure has line symmetry if the figure can be constructed. So there are 6 lines of symmetry. mapped onto itself by a reflection in a line. This

Next, rotate the square about the center point. The figure has 4 lines of symmetry. image maps to the original at 60, 120, 180, 240, 300, and 360 degree rotations. So the order of symmetry is 6.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure also has rotational symmetry.

Therefore, neither of them are correct. Figure A has both line and rotational symmetry. d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines ANSWER: of symmetry and the order of symmetry. A regular Neither; Figure A has both line and rotational polygon with n sides has n lines of symmetry and symmetry. order of symmetry n. 42. CHALLENGE A quadrilateral in the coordinate ANSWER: plane has exactly two lines of symmetry, y = x – 1 a. 3 and y = –x + 2. Find a set of possible vertices for b. 3 the figure. Graph the figure and the lines of c. symmetry. SOLUTION: Graph the figure and the lines of symmetry.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

Pick points that are the same distance a from one line and the same distance b from the other line. In the same answer, the quadrilateral is a rectangle with sides which are parallel to the lines of symmetry. This guarantees that the vertices of the quadrilateral SOLUTION: are the same distance a from one line and the same A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. This distance b from the other line. In this case, a = figure has 4 lines of symmetry. and b = .

A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2).

ANSWER: Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2)

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure also has rotational symmetry. 43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. Therefore, neither of them are correct. Figure A has SOLUTION: both line and rotational symmetry. circle; Every line through the center of a circle is a ANSWER: line of symmetry, and there are infinitely many such Neither; Figure A has both line and rotational lines. symmetry. ANSWER: 42. CHALLENGE A quadrilateral in the coordinate circle; Every line through the center of a circle is a plane has exactly two lines of symmetry, y = x – 1 line of symmetry, and there are infinitely many such lines. and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of 44. OPEN-ENDED Draw a figure with line symmetry symmetry. but not rotational symmetry. Explain. SOLUTION: SOLUTION: Graph the figure and the lines of symmetry. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not have rotational symmetry.

An isosceles triangle has line symmetry from the Pick points that are the same distance a from one vertex angle to the base of the triangle, but it does line and the same distance b from the other line. In not have rotational symmetry because it cannot be the same answer, the quadrilateral is a rectangle with rotated from 0° to 360° and map onto itself. sides which are parallel to the lines of symmetry. This guarantees that the vertices of the quadrilateral are the same distance a from one line and the same distance b from the other line. In this case, a =

and b = . ANSWER: Sample answer: An isosceles triangle has line A set of possible vertices for the figure are, (–1, 0), symmetry from the vertex angle to the base of the (2, 3), (4, 1), and (1, 2). triangle, but it does not have rotational symmetry ANSWER: because it cannot be rotated from 0° to 360° and map onto itself. Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2)

45. WRITING IN MATH How are line symmetry and rotational symmetry related?

SOLUTION: In both types of symmetries the figure is mapped 43. REASONING A figure has infinitely many lines of onto itself. symmetry. What is the figure? Explain. Rotational symmetry. SOLUTION: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines. Reflectional symmetry:

ANSWER: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines.

44. OPEN-ENDED Draw a figure with line symmetry In some cases an object can have both rotational and but not rotational symmetry. Explain. reflectional symmetry, such as the diamond, however SOLUTION: some objects do not have both such as the crab. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not

have rotational symmetry. ANSWER: An isosceles triangle has line symmetry from the Sample answer: In both rotational and line symmetry vertex angle to the base of the triangle, but it does a figure is mapped onto itself. However, in line not have rotational symmetry because it cannot be symmetry the figure is mapped onto itself by a rotated from 0° to 360° and map onto itself. reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry.

46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of symmetry and the order of symmetry, and then she ANSWER: enters this value into a database. Which value should Sample answer: An isosceles triangle has line she enter in the database for the tile shown here? symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

A 2 B 3 C 4 WRITING IN MATH 45. How are line symmetry and D 8 rotational symmetry related? SOLUTION: SOLUTION: In both types of symmetries the figure is mapped onto itself.

Rotational symmetry.

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile. Reflectional symmetry: It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn around its center.

2 + 2 = 4, so C is the correct answer.

ANSWER: In some cases an object can have both rotational and C reflectional symmetry, such as the diamond, however some objects do not have both such as the crab. 47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A

ANSWER:

Sample answer: In both rotational and line symmetry C a figure is mapped onto itself. However, in line symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have D line symmetry and rotational symmetry.

46. Sasha owns a tile store. For each tile in her store, she E calculates the sum of the number of lines of symmetry and the order of symmetry, and then she enters this value into a database. Which value should she enter in the database for the tile shown here? SOLUTION:

Option A has rotational and reflectional symmetry.

Option B has reflectional symmetry but not rotational A 2 symmetry. B 3 C 4 D 8 SOLUTION:

Option C has neither rotational nor reflectional symmetry.

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile. Option D has rotational symmetry but not reflectional symmetry. It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn around its center.

2 + 2 = 4, so C is the correct answer. Option E has reflectional symmetry but not rotational ANSWER: symmetry. C

47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A

The correct choice is D.

B ANSWER: D

C 48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle D C Isosceles triangle D Scalene triangle

E SOLUTION: An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

SOLUTION: Option A has rotational and reflectional symmetry. ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS Option B has reflectional symmetry but not rotational has line symmetry but not rotational symmetry? symmetry. A B C D SOLUTION: Option C has neither rotational nor reflectional First, plot the points. symmetry.

Option D has rotational symmetry but not reflectional symmetry.

Option E has reflectional symmetry but not rotational symmetry. Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

The correct choice is D.

ANSWER: D

48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle

D Scalene triangle Option B has reflective symmetry but not rotational SOLUTION: symmetry. The correct choice is B. An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D SOLUTION: First, plot the points.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

Option B has reflective symmetry but not rotational symmetry. The correct choice is B. State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

Thus, there are four possible lines that go through the center and are lines of reflections.

State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number. Therefore, the figure has four lines of symmetry. ANSWER: yes; 4

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

2. Two lines of reflection go through the sides of the figure. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER: Two lines of reflection go through the vertices of the no figure.

3. SOLUTION:

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. Thus, there are four possible lines that go through

the center and are lines of reflections. The given figure has reflectional symmetry.

The figure has a vertical line of symmetry.

Therefore, the figure has four lines of symmetry. It does not have a horizontal line of symmetry. ANSWER: yes; 4

The figure does not have a line of symmetry through the vertices.

2. SOLUTION: Thus, the figure has only one line of symmetry. A figure has reflectional symmetry if the figure can ANSWER: be mapped onto itself by a reflection in a line. yes; 1 The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

3-5 SymmetryANSWER: no State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

4. The given figure has reflectional symmetry. SOLUTION: A figure in the plane has rotational symmetry if the The figure has a vertical line of symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: It does not have a horizontal line of symmetry. no

The figure does not have a line of symmetry through 5. the vertices. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

Thus, the figure has only one line of symmetry.

ANSWER: yes; 1

The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of symmetry. State whether the figure has rotational symmetry. Write yes or no. If so, copy the The given figure has order of symmetry of 2, since figure, locate the center of symmetry, and state the figure can be rotated twice in 360°. the order and magnitude of symmetry. eSolutions Manual - Powered by Cognero The magnitude of symmetry is the smallest anglePage 2 through which a figure can be rotated so that it maps onto itself. 4. Since the figure has order 2 rotational symmetry, the SOLUTION: magnitude of the symmetry is . A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation ANSWER: between 0° and 360° about the center of the figure. yes; 2; 180°

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: no

6. SOLUTION: 5. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The given figure has rotational symmetry. between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

Since the figure can be rotated 4 times within 360° , The number of times a figure maps onto itself as it it has order 4 rotational symmetry rotates form 0° and 360° is called the order of symmetry. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The given figure has order of symmetry of 2, since onto itself.

the figure can be rotated twice in 360°. The figure has magnitude of symmetry of

. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps ANSWER: onto itself. yes; 4; 90°

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is .

ANSWER: yes; 2; 180°

State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

6. SOLUTION: 7. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. Vertical and horizontal lines through the center and diagonal lines through two opposite vertices are all The given figure has rotational symmetry. lines of symmetry for a square oriented this way.

The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1.

Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

The number of times a figure maps onto itself as it ANSWER: rotates from 0° to 360° is called the order of line symmetry; rotational symmetry; the reflection in symmetry. the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in Since the figure can be rotated 4 times within 360° , the line y = -x - 1 map the square onto itself; the it has order 4 rotational symmetry rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER: yes; 4; 90° 8. SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself. State whether the figure has line symmetry and/or rotational symmetry. If so, describe the ANSWER: reflections and/or rotations that map the figure rotational symmetry; the rotation of 180 degrees onto itself. around the point (1, 1) maps the parallelogram onto itself.

REGULARITY State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number. 7. SOLUTION: Vertical and horizontal lines through the center and 9. diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way. SOLUTION: A figure has reflectional symmetry if the figure can The equations of those lines in this figure are x = 0, be mapped onto itself by a reflection in a line. y = -1, y = x - 1, and y = -x - 1. For the given figure, there are no lines of reflection Each quarter turn also maps the square onto itself. where the figure can map onto itself. Thus, the figure So the rotations of 90, 180, and 270 degrees around does not have any lines of of symmetry. the point (0, -1) map the square onto itself. ANSWER: ANSWER: no line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself. 10. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

8. In order for the figure to map onto itself, the line of reflection must go through the center point. SOLUTION: This figure does not have line symmetry, because The figure has a vertical and horizontal line of adjacent sides are not congruent. reflection.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

ANSWER:

rotational symmetry; the rotation of 180 degrees around the point (1, 1) maps the parallelogram onto It is also possible to have reflection over the itself. diagonal lines. REGULARITY State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

Therefore, the figure has four lines of symmetry 9. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry. ANSWER: ANSWER: yes; 4 no

10.

SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry. 11. In order for the figure to map onto itself, the line of SOLUTION: reflection must go through the center point. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The figure has a vertical and horizontal line of reflection. The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though opposites edges. It is also possible to have reflection over the diagonal lines.

There are three lines of reflection that go though opposites vertices. Therefore, the figure has four lines of symmetry

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has

ANSWER: six lines of symmetry. yes; 4

ANSWER: yes; 6

11. SOLUTION: A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line 12. of reflection must go through the center point. SOLUTION: A figure has reflectional symmetry if the figure can There are three lines of reflection that go though be mapped onto itself by a reflection in a line. opposites edges. The figure has reflectional symmetry.

There is only one line of symmetry, a horizontal line through the middle of the figure.

There are three lines of reflection that go though opposites vertices. Thus, the figure has one line of symmetry.

ANSWER: yes; 1

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry.

13. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

ANSWER: There is only one possible line of reflection, yes; 6 horizontally though the middle of the figure.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

12. SOLUTION: A figure has reflectional symmetry if the figure can 14. be mapped onto itself by a reflection in a line. SOLUTION: The figure has reflectional symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. There is only one line of symmetry, a horizontal line through the middle of the figure. The given figure does not have reflectional symmetry. It is not possible to draw a line of reflection where the figure can map onto itself.

ANSWER: no Thus, the figure has one line of symmetry. FLAGS State whether each flag design appears to ANSWER: have line symmetry. Write yes or no. If so, copy the yes; 1 flag, draw all lines of symmetry, and state their number. 15. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The flag does not have any reflectional symmetry. If 13. the red lines in the diagonals were in the same SOLUTION: location above and below the center horizontal line, A figure has reflectional symmetry if the figure can the flag would have three lines of symmetry. be mapped onto itself by a reflection in a line. ANSWER:

The figure has reflectional symmetry. no

There is only one possible line of reflection, 16. Refer to the flag on page 262. horizontally though the middle of the figure. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

Thus, the figure has one line of symmetry. The figure has reflectional symmetry.

ANSWER: In order for the figure to map onto itself, the line of yes; 1 reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

14. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. It is not possible to draw a line of Two diagonal lines of reflection are possible. reflection where the figure can map onto itself.

ANSWER: no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their number. 15. Refer to page 262. SOLUTION: There are a total of four possible lines that go A figure has reflectional symmetry if the figure can through the center and are lines of reflections. Thus,

be mapped onto itself by a reflection in a line. the flag has four lines of symmetry.

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER: no

16. Refer to the flag on page 262. SOLUTION: ANSWER: A figure has reflectional symmetry if the figure can yes; 4 be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

17. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag. Two diagonal lines of reflection are possible.

It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry.

There are a total of four possible lines that go ANSWER: through the center and are lines of reflections. Thus, yes; 1 the flag has four lines of symmetry.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

ANSWER: yes; 4 18. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

17. Refer to page 262. The figure has rotational symmetry. SOLUTION: A figure has reflectional symmetry if the figure can The number of times a figure maps onto itself as it be mapped onto itself by a reflection in a line. rotates from 0° to 360° is called the order of symmetry. The figure has reflectional symmetry. This figure has order 2 rotational symmetry, since A horizontal line is a line of reflections for this flag. you have to rotate 180° to get the figure to map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

It is not possible to reflect over a vertical or line The figure has a magnitude of symmetry of through the diagonals. .

Thus, the figure has one line of symmetry. ANSWER:

ANSWER: yes; 1

yes; 2; 180°

State whether the figure has rotational symmetry. Write yes or no. If so, copy the 19. figure, locate the center of symmetry, and state SOLUTION: the order and magnitude of symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

18. The triangle has rotational symmetry. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The figure has rotational symmetry. symmetry.

The number of times a figure maps onto itself as it The figure has order 3 rotational symmetry. rotates from 0° to 360° is called the order of symmetry. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps This figure has order 2 rotational symmetry, since onto itself. you have to rotate 180° to get the figure to map onto itself. The figure has magnitude of symmetry of . The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps ANSWER: onto itself.

The figure has a magnitude of symmetry of .

ANSWER: yes; 3; 120°

20. yes; 2; 180° SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

19. The isosceles trapezoid has no rotational symmetry. SOLUTION: There is no way to rotate it such that it can be mapped onto itself. A figure in the plane has rotational symmetry if the

figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. ANSWER:

no The triangle has rotational symmetry.

21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The number of times a figure maps onto itself as it between 0° and 360° about the center of the figure. rotates from 0° to 360° is called the order of symmetry. The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it The figure has order 3 rotational symmetry. can be mapped onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 3; 120°

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. ANSWER: The isosceles trapezoid has no rotational symmetry. no There is no way to rotate it such that it can be mapped onto itself.

ANSWER: no 22. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. 21. The figure has rotational symmetry. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 8; 45° ANSWER: no

23. SOLUTION: 22. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The figure has rotational symmetry. between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This The number of times a figure maps onto itself as it means that the figure can be rotated 8 times and map rotates from 0° to 360° is called the order of onto itself within 360°. symmetry. The magnitude of symmetry is the smallest angle The figure has order 8 rotational symmetry. This through which a figure can be rotated so that it maps implies you can rotate the figure 8 times and have it onto itself. map onto itself within 360°. The figure has magnitude of symmetry of The magnitude of symmetry is the smallest angle . through which a figure can be rotated so that it maps onto itself. ANSWER:

The figure has magnitude of symmetry of .

ANSWER: yes; 8; 45°

WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, state the order and magnitude of symmetry. 24. Refer to page 263.

yes; 8; 45° SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The wheel has rotational symmetry. 23.

SOLUTION: The number of times a figure maps onto itself as it A figure in the plane has rotational symmetry if the rotates from 0° to 360° is called the order of figure can be mapped onto itself by a rotation symmetry. between 0° and 360° about the center of the figure. The wheel has order 5 rotational symmetry. There The figure has rotational symmetry. are 5 large spokes and 5 small spokes. You can rotate the wheel 5 times within 360° and map the figure onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The number of times a figure maps onto itself as it The wheel has magnitude of symmetry rotates from 0° to 360° is called the order of . symmetry. ANSWER: The figure has order 8 rotational symmetry. This yes; 5; 72° means that the figure can be rotated 8 times and map onto itself within 360°. 25. Refer to page 263. The magnitude of symmetry is the smallest angle SOLUTION: through which a figure can be rotated so that it maps A figure in the plane has rotational symmetry if the onto itself. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure has magnitude of symmetry of . The wheel has rotational symmetry.

ANSWER: The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The wheel has order 8 rotational symmetry. There yes; 8; 45° are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself. WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, The magnitude of symmetry is the smallest angle state the order and magnitude of symmetry. through which a figure can be rotated so that it maps 24. Refer to page 263. onto itself.

SOLUTION: The wheel has order 8 rotational symmetry and A figure in the plane has rotational symmetry if the magnitude . figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. ANSWER:

yes; 8; 45° The wheel has rotational symmetry. 26. Refer to page 263. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of SOLUTION: symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The wheel has order 5 rotational symmetry. There between 0° and 360° about the center of the figure. are 5 large spokes and 5 small spokes. You can The wheel has rotational symmetry. rotate the wheel 5 times within 360° and map the figure onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The magnitude of symmetry is the smallest angle symmetry. The wheel has order 10 rotational through which a figure can be rotated so that it maps symmetry. There are 10 bolts and the tire can be onto itself. rotated 10 times within 360° and map onto itself.

The wheel has magnitude of symmetry The magnitude of symmetry is the smallest angle . through which a figure can be rotated so that it maps onto itself. The wheel has magnitude of symmetry of ANSWER: . yes; 5; 72° ANSWER: yes; 10; 36° 25. Refer to page 263. State whether the figure has line symmetry SOLUTION: and/or rotational symmetry. If so, describe the A figure in the plane has rotational symmetry if the reflections and/or rotations that map the figure figure can be mapped onto itself by a rotation onto itself. between 0° and 360° about the center of the figure.

The wheel has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. 27. The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times SOLUTION: within 360° and map onto itself. This triangle is scalene, so it cannot have symmetry.

The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps no symmetry onto itself.

The wheel has order 8 rotational symmetry and magnitude .

ANSWER: yes; 8; 45° 28. 26. Refer to page 263. SOLUTION: SOLUTION: This figure is a square, because each pair of adjacent A figure in the plane has rotational symmetry if the sides is congruent and perpendicular. figure can be mapped onto itself by a rotation All squares have both line and rotational symmetry. between 0° and 360° about the center of the figure. The line symmetry is vertically, horizontally, and The wheel has rotational symmetry. diagonally through the center of the square, with lines that are either parallel to the sides of the square or The number of times a figure maps onto itself as it that include two vertices of the square. The rotates from 0° to 360° is called the order of equations of those lines are: x = 0, y = 0, y = x, and y symmetry. The wheel has order 10 rotational = -x symmetry. There are 10 bolts and the tire can be rotated 10 times within 360° and map onto itself. The rotational symmetry is for each quarter turn in a square, so the rotations of 90, 180, and 270 degrees The magnitude of symmetry is the smallest angle around the origin map the square onto itself. through which a figure can be rotated so that it maps onto itself. The wheel has magnitude of symmetry of . ANSWER: line symmetry; rotational symmetry; the reflection in ANSWER: the line x = 0, the reflection in the line y = 0, the yes; 10; 36° reflection in the line y = x, and the reflection in the line y = -x all map the square onto itself; the rotations State whether the figure has line symmetry of 90, 180, and 270 degrees around the origin map and/or rotational symmetry. If so, describe the the square onto itself. reflections and/or rotations that map the figure onto itself.

29. SOLUTION: 27. The trapezoid has line symmetry, because it is SOLUTION: isosceles, but it does not have rotational symmetry, This triangle is scalene, so it cannot have symmetry. because no trapezoid does.

ANSWER: The reflection in the line y = 1.5 maps the trapezoid no symmetry onto itself, because that is the perpendicular bisector to the parallel sides.

ANSWER: line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself.

28. SOLUTION: This figure is a square, because each pair of adjacent sides is congruent and perpendicular. All squares have both line and rotational symmetry. 30. The line symmetry is vertically, horizontally, and diagonally through the center of the square, with lines SOLUTION: that are either parallel to the sides of the square or This figure is a parallelogram, so it has rotational that include two vertices of the square. The symmetry of a half turn or 180 degrees around its equations of those lines are: x = 0, y = 0, y = x, and y center, which is the point (1, -1.5). = -x Since this parallelogram is not a rhombus it does not The rotational symmetry is for each quarter turn in a have line symmetry. square, so the rotations of 90, 180, and 270 degrees around the origin map the square onto itself. ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram ANSWER: onto itself. line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = 0, the 31. MODELING Symmetry is an important component reflection in the line y = x, and the reflection in the of photography. Photographers often use reflection in line y = -x all map the square onto itself; the rotations water to create symmetry in photos. The photo on of 90, 180, and 270 degrees around the origin map page 263 is a long exposure shot of the Eiffel tower the square onto itself. reflected in a pool.

a. Describe the two-dimensional symmetry created by the photo. b. Is there rotational symmetry in the photo? Explain your reasoning. SOLUTION: 29. a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. SOLUTION: There is a vertical line of symmetry through the The trapezoid has line symmetry, because it is center of the photo. isosceles, but it does not have rotational symmetry, b No; sample answer: Because of how the image is because no trapezoid does. reflected over the horizontal line, there is no rotational symmetry. The reflection in the line y = 1.5 maps the trapezoid onto itself, because that is the perpendicular bisector ANSWER: to the parallel sides. a. Sample answer: There is a horizontal line of ANSWER: symmetry between the tower and its reflection. There is a vertical line of symmetry through the line symmetry; the reflection in the line y = 1.5 maps center of the photo. the trapezoid onto itself. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no rotational symmetry.

COORDINATE GEOMETRY Determine whether the figure with the given vertices has line symmetry and/or rotational symmetry. 32. R(–3, 3), S(–3, –3), T(3, 3) 30. SOLUTION: SOLUTION: Draw the figure on a coordinate plane. This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its center, which is the point (1, -1.5).

Since this parallelogram is not a rhombus it does not have line symmetry.

ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram onto itself. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. 31. MODELING Symmetry is an important component of photography. Photographers often use reflection in The given triangle has a line of symmetry through water to create symmetry in photos. The photo on points (0, 0) and (–3, 3). page 263 is a long exposure shot of the Eiffel tower reflected in a pool. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation a. Describe the two-dimensional symmetry created between 0° and 360° about the center of the figure. by the photo. There is not way to rotate the figure and have it map b. Is there rotational symmetry in the photo? Explain onto itself. your reasoning. Thus, the figure has only line symmetry. SOLUTION: a Sample answer: There is a horizontal line of ANSWER: symmetry between the tower and its reflection. line There is a vertical line of symmetry through the center of the photo. 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) b No; sample answer: Because of how the image is reflected over the horizontal line, there is no SOLUTION: rotational symmetry. Draw the figure on a coordinate plane. ANSWER: a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no

rotational symmetry. COORDINATE GEOMETRY Determine A figure has line symmetry if the figure can be whether the figure with the given vertices has line mapped onto itself by a reflection in a line. The symmetry and/or rotational symmetry. given figure has 4 lines of symmetry. The line of 32. R(–3, 3), S(–3, –3), T(3, 3) symmetry are though the following pairs of points {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, SOLUTION: and {(2, 2), (2, –2)}. Draw the figure on a coordinate plane. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4.

Thus, the figure has both line symmetry and rotational symmetry.

A figure has line symmetry if the figure can be ANSWER: mapped onto itself by a reflection in a line. line and rotational

34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – The given triangle has a line of symmetry through 2) points (0, 0) and (–3, 3). SOLUTION: A figure in the plane has rotational symmetry if the Draw the figure on a coordinate plane. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. There is not way to rotate the figure and have it map onto itself.

Thus, the figure has only line symmetry.

ANSWER: line A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) given hexagon has 2 lines of symmetry. The lines SOLUTION: pass through the following pair of points {(0, 4), (0, – 4)}, and {(3, 0), (–3, 0)} Draw the figure on a coordinate plane. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure has rotational symmetry. You can rotate the figure once within 360° and have it map to itself.

Thus, the figure has both line symmetry and rotational symmetry.

A figure has line symmetry if the figure can be ANSWER: mapped onto itself by a reflection in a line. The line and rotational given figure has 4 lines of symmetry. The line of symmetry are though the following pairs of points 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, and {(2, 2), (2, –2)}. SOLUTION: Draw the figure on a coordinate plane. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: A figure has line symmetry if the figure can be line and rotational mapped onto itself by a reflection in a line. The trapezoid has a line of reflection through points (0,3) 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – and (0, –3). 2) SOLUTION: A figure in the plane has rotational symmetry if the Draw the figure on a coordinate plane. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. There is no way to rotate this figure and have it map onto itself. Thus, it does not have rotational symmetry.

Therefore, the figure has only line symmetry.

ANSWER: line A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The ALGEBRA Graph the function and determine given hexagon has 2 lines of symmetry. The lines whether the graph has line and/or rotational pass through the following pair of points {(0, 4), (0, – symmetry. If so, state the order and magnitude of 4)}, and {(3, 0), (–3, 0)} symmetry, and write the equations of any lines of symmetry. A figure in the plane has rotational symmetry if the 36. y = x figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. Graph the function. The figure has rotational symmetry. You can rotate the figure once within 360° and have it map to itself.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3)

SOLUTION: A figure has reflectional symmetry if the figure can Draw the figure on a coordinate plane. be mapped onto itself by a reflection in a line. The line y = x has reflectional symmetry since any line perpendicular to y = x is a line of reflection. The equation of the line symmetry is y = –x.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The line can be rotated twice within 360° and be mapped onto itself. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The The number of times a figure maps onto itself as it trapezoid has a line of reflection through points (0,3) rotates from 0° to 360° is called the order of and (0, –3). symmetry. The graph has order 2 rotational symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The magnitude of symmetry is the smallest angle between 0° and 360° about the center of the through which a figure can be rotated so that it maps figure. There is no way to rotate this figure and have onto itself. it map onto itself. Thus, it does not have rotational The graph has magnitude of symmetry of symmetry. .

Therefore, the figure has only line symmetry. Thus, the graph has both reflectional and rotational ANSWER: symmetry. line ANSWER: ALGEBRA Graph the function and determine rotational; 2; 180°; line symmetry; y = –x whether the graph has line and/or rotational symmetry. If so, state the order and magnitude of symmetry, and write the equations of any lines of symmetry. 36. y = x SOLUTION: Graph the function.

2 37. y = x + 1 SOLUTION: Graph the function.

A figure in the plane has rotational symmetry if the A figure has reflectional symmetry if the figure can figure can be mapped onto itself by a rotation be mapped onto itself by a reflection in a line. The between 0° and 360° about the center of the figure. graph is reflected through the y-axis. Thus, the The line can be rotated twice within 360° and be equation of the line symmetry is x = 0. mapped onto itself. A figure in the plane has rotational symmetry if the The number of times a figure maps onto itself as it figure can be mapped onto itself by a rotation rotates from 0° to 360° is called the order of between 0° and 360° about the center of the symmetry. The graph has order 2 rotational figure. There is no way to rotate the graph and have symmetry. it map onto itself.

The magnitude of symmetry is the smallest angle Thus, the graph has only reflectional symmetry. through which a figure can be rotated so that it maps onto itself. ANSWER: The graph has magnitude of symmetry of line; x = 0 .

Thus, the graph has both reflectional and rotational symmetry.

ANSWER: rotational; 2; 180°; line symmetry; y = –x

38. y = –x3 SOLUTION: Graph the function.

2 37. y = x + 1 SOLUTION: Graph the function.

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself.

it map onto itself. through which a figure can be rotated so that it maps

onto itself. The graph has magnitude of symmetry of Thus, the graph has only reflectional symmetry. . ANSWER: line; x = 0 Thus, the graph has only rotational symmetry. ANSWER: rotational; 2; 180°

38. y = –x3

SOLUTION: Graph the function. 39. Refer to the rectangle on the coordinate plane.

a. What are the equations of the lines of symmetry of the rectangle? A figure has reflectional symmetry if the figure can b. What happens to the equations of the lines of be mapped onto itself by a reflection in a line. The symmetry when the rectangle is rotated 90 degrees graph does not have a line of reflections where the counterclockwise around its center of symmetry? graph can be mapped onto itself. Explain.

A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation a. The lines of symmetry are parallel to the sides of between 0° and 360° about the center of the the rectangles, and through the center of rotation. figure. You can rotate the graph through the origin and have it map onto itself. The slopes of the sides of the rectangle are 0.5 and -2, so the slopes of the lines of symmetry are the The number of times a figure maps onto itself as it same. rotates from 0° to 360° is called the order of The center of the rectangle is (1, 1.5). Use the symmetry. The graph has order 2 rotational point-slope formula to find equations. symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of b. The equations of the lines of symmetry do not . change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are Thus, the graph has only rotational symmetry. mapped to each other.

ANSWER: ANSWER: rotational; 2; 180° a. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

39. Refer to the rectangle on the coordinate plane. a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. a. What are the equations of the lines of symmetry of c. Tabular Repeat the process in parts a and b for a the rectangle? square, regular pentagon, and regular hexagon. b. What happens to the equations of the lines of Record the number of lines of symmetry and the symmetry when the rectangle is rotated 90 degrees order of symmetry for each polygon. counterclockwise around its center of symmetry? d. Verbal Make a conjecture about the number of Explain. lines of symmetry and the order of symmetry for a regular polygon with n sides. SOLUTION: SOLUTION: a. The lines of symmetry are parallel to the sides of a. the rectangles, and through the center of rotation. Construct an equilateral triangle and label the vertices A, B, and C. Draw a line through A

perpendicular to . Reflect the triangle in the line. The slopes of the sides of the rectangle are 0.5 and Show the labels of the reflected image. If the image -2, so the slopes of the lines of symmetry are the maps to the original, then this line is a line of same. reflection. The center of the rectangle is (1, 1.5). Use the point-slope formula to find equations.

b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

ANSWER: Next, draw a line through B perpendicular to . a. Reflect the triangle in the line. Show the labels of the b. The equations of the lines of symmetry do not reflected image. If the image maps to the original, change; although the rectangle does not map onto then this line is a line of reflection. itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool Lastly, draw a line through C perpendicular to . under the transformation menu to investigate and Reflect the triangle in the line. Show the labels of the determine all possible lines of symmetry. Then record reflected image. If the image maps to the original, their number. then this line is a line of reflection. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides. SOLUTION: a. Construct an equilateral triangle and label the vertices A, B, and C. Draw a line through A perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of There are 3 lines of symmetry. reflection. b. Construct an equilateral triangle and show the labels of the vertices. Next, find the center of the triangle. Since this is an equilateral triangle, the circumcenter, incenter, centroid, and orthocenter are the same point. Construct altitudes through each vertex and label the intersection. Rotate the triangle about point D. A 120 degree rotation will map the image to the original. Show the labels of the image.

Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Lastly, draw a line through C perpendicular to . Show the labels of the image Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

There are 3 lines of symmetry.

b. Construct an equilateral triangle and show the labels of the vertices. Next, find the center of the triangle. Since this is an equilateral triangle, the circumcenter, incenter, centroid, and orthocenter are the same point. Construct altitudes through each vertex and label the intersection. Rotate the triangle about point D. A 120 degree rotation will map the image to the original. Show the Since the figure maps onto itself 3 times as it is labels of the image. rotated, the order of symmetry is 3. c. Square Construct a square and then construct lines through the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps The triangle can be rotated a third time about D. A onto the original when reflected in each of the 5 lines 360 degree rotation maps the image to the original. constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5.

Since the figure maps onto itself 3 times as it is rotated, the order of symmetry is 3. c. Square

Construct a square and then construct lines through Regular Hexagon the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto Construct a regular hexagon and then construct lines the original when reflected in each of the 4 lines through each vertex perpendicular to the sides. Use constructed. So there are 4 lines of symmetry. the reflection tool first to find that the image maps Next, rotate the square about the center point. The onto the original when reflected in each of the 6 lines image maps to the original at 90, 180, 270, and 360 constructed. So there are 6 lines of symmetry. degree rotations. So the order of symmetry is 4. Next, rotate the square about the center point. The image maps to the original at 60, 120, 180, 240, 300, and 360 degree rotations. So the order of symmetry is 6.

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. d. Sample answer: for each figure studied, the Next, rotate the square about the center point. The number of sides of the figure is the same as the lines image maps to the original at 72, 144, 216, 288, and of symmetry and the order of symmetry. A regular 360 degree rotations. So the order of symmetry is 5. polygon with n sides has n lines of symmetry and order of symmetry n.

ANSWER: a. 3 b. 3 c.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A

has only rotational symmetry. Is either of them Regular Hexagon correct? Explain your reasoning. Construct a regular hexagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 6 lines constructed. So there are 6 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 60, 120, 180, 240, 300, SOLUTION: and 360 degree rotations. So the order of symmetry A figure has line symmetry if the figure can be is 6. mapped onto itself by a reflection in a line. This figure has 4 lines of symmetry.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines The figure also has rotational symmetry. of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and Therefore, neither of them are correct. Figure A has order of symmetry n. both line and rotational symmetry.

ANSWER: ANSWER: a. 3 Neither; Figure A has both line and rotational b. 3 symmetry. c. 42. CHALLENGE A quadrilateral in the coordinate plane has exactly two lines of symmetry, y = x – 1 and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of d. Sample answer: A regular polygon with n sides symmetry. has n lines of symmetry and order of symmetry n. SOLUTION: 41. ERROR ANALYSIS Jaime says that Figure A has Graph the figure and the lines of symmetry. only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

SOLUTION: Pick points that are the same distance a from one A figure has line symmetry if the figure can be line and the same distance b from the other line. In mapped onto itself by a reflection in a line. This the same answer, the quadrilateral is a rectangle with figure has 4 lines of symmetry. sides which are parallel to the lines of symmetry. This guarantees that the vertices of the quadrilateral are the same distance a from one line and the same distance b from the other line. In this case, a =

and b = .

A set of possible vertices for the figure are, (–1, 0), A figure in the plane has rotational symmetry if the (2, 3), (4, 1), and (1, 2). figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. ANSWER: Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2)

The figure also has rotational symmetry.

Therefore, neither of them are correct. Figure A has both line and rotational symmetry.

ANSWER: 43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. Neither; Figure A has both line and rotational symmetry. SOLUTION: circle; Every line through the center of a circle is a 42. CHALLENGE A quadrilateral in the coordinate line of symmetry, and there are infinitely many such plane has exactly two lines of symmetry, y = x – 1 lines. and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of ANSWER: symmetry. circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such SOLUTION: lines. Graph the figure and the lines of symmetry. 44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Pick points that are the same distance a from one Identify a figure that has line symmetry but does not line and the same distance b from the other line. In have rotational symmetry. the same answer, the quadrilateral is a rectangle with sides which are parallel to the lines of symmetry. This guarantees that the vertices of the quadrilateral An isosceles triangle has line symmetry from the are the same distance a from one line and the same vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be distance b from the other line. In this case, a = rotated from 0° to 360° and map onto itself.

and b = .

A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2). ANSWER: ANSWER: Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) Sample answer: An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

43. REASONING A figure has infinitely many lines of 45. WRITING IN MATH How are line symmetry and symmetry. What is the figure? Explain. rotational symmetry related? SOLUTION: SOLUTION: circle; Every line through the center of a circle is a In both types of symmetries the figure is mapped line of symmetry, and there are infinitely many such onto itself. lines. Rotational symmetry. ANSWER: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such Reflectional symmetry: lines.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure In some cases an object can have both rotational and in the plane has rotational symmetry if the figure can reflectional symmetry, such as the diamond, however be mapped onto itself by a rotation between 0° and some objects do not have both such as the crab. 360° about the center of the figure.

Identify a figure that has line symmetry but does not have rotational symmetry.

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself. ANSWER: Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have ANSWER: line symmetry and rotational symmetry. Sample answer: An isosceles triangle has line symmetry from the vertex angle to the base of the 46. Sasha owns a tile store. For each tile in her store, she triangle, but it does not have rotational symmetry calculates the sum of the number of lines of because it cannot be rotated from 0° to 360° and symmetry and the order of symmetry, and then she map onto itself. enters this value into a database. Which value should she enter in the database for the tile shown here?

45. WRITING IN MATH How are line symmetry and rotational symmetry related? A 2 SOLUTION: B 3 In both types of symmetries the figure is mapped C 4 onto itself. D 8

Rotational symmetry. SOLUTION:

Reflectional symmetry:

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn In some cases an object can have both rotational and around its center. reflectional symmetry, such as the diamond, however some objects do not have both such as the crab. 2 + 2 = 4, so C is the correct answer.

ANSWER: C

47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A ANSWER: Sample answer: In both rotational and line symmetry

a figure is mapped onto itself. However, in line B symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry. C

46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of

symmetry and the order of symmetry, and then she D enters this value into a database. Which value should she enter in the database for the tile shown here? E

SOLUTION: Option A has rotational and reflectional symmetry.

A 2 B 3 C 4

D 8 Option B has reflectional symmetry but not rotational SOLUTION: symmetry.

The tile is a rhombus and has 2 lines of symmetry. Option C has neither rotational nor reflectional Each connects opposite corners of the tile. symmetry.

It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn around its center. Option D has rotational symmetry but not reflectional 2 + 2 = 4, so C is the correct answer. symmetry.

ANSWER: C

47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could Option E has reflectional symmetry but not rotational be the figure that Patrick drew? symmetry. A

C The correct choice is D. ANSWER: D D 48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? E A Equilateral triangle B Equiangular triangle C Isosceles triangle D Scalene triangle SOLUTION: SOLUTION: Option A has rotational and reflectional symmetry. An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

ANSWER:

Option B has reflectional symmetry but not rotational C symmetry. 49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A Option C has neither rotational nor reflectional B symmetry. C D SOLUTION: First, plot the points. Option D has rotational symmetry but not reflectional symmetry.

Option E has reflectional symmetry but not rotational symmetry.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational The correct choice is D. symmetry.

ANSWER: D

48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle D Scalene triangle SOLUTION: An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C. Option B has reflective symmetry but not rotational ANSWER: symmetry. The correct choice is B. C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D SOLUTION: First, plot the points.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

Thus, there are four possible lines that go through the center and are lines of reflections.

State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

Therefore, the figure has four lines of symmetry.

ANSWER: 1. yes; 4 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure. 2. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto Two lines of reflection go through the vertices of the itself. figure. ANSWER: no

3. Thus, there are four possible lines that go through SOLUTION: the center and are lines of reflections. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

The figure has a vertical line of symmetry.

Therefore, the figure has four lines of symmetry.

ANSWER: yes; 4 It does not have a horizontal line of symmetry.

The figure does not have a line of symmetry through the vertices.

2. SOLUTION:

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. Thus, the figure has only one line of symmetry.

The given figure does not have reflectional ANSWER: symmetry. There is no way to fold or reflect it onto yes; 1 itself.

ANSWER: no

State whether the figure has rotational 3. symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state SOLUTION: the order and magnitude of symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry. 4. The figure has a vertical line of symmetry. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational It does not have a horizontal line of symmetry. symmetry.

ANSWER: no

The figure does not have a line of symmetry through the vertices.

5. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Thus, the figure has only one line of symmetry. The given figure has rotational symmetry.

ANSWER: yes; 1

State whether the figure has rotational The number of times a figure maps onto itself as it symmetry. Write yes or no. If so, copy the rotates form 0° and 360° is called the order of figure, locate the center of symmetry, and state symmetry. the order and magnitude of symmetry. The given figure has order of symmetry of 2, since the figure can be rotated twice in 360°.

The magnitude of symmetry is the smallest angle

4. through which a figure can be rotated so that it maps SOLUTION: onto itself. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation Since the figure has order 2 rotational symmetry, the between 0° and 360° about the center of the figure. magnitude of the symmetry is .

For the given figure, there is no rotation between 0° ANSWER: and 360° that maps the figure onto itself. If the figure yes; 2; 180° were a regular pentagon, it would have rotational symmetry.

ANSWER: 3-5 Symmetryno

5. 6. SOLUTION: SOLUTION: A figure in the plane has rotational symmetry if the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. between 0° and 360° about the center of the figure.

The given figure has rotational symmetry. The given figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The number of times a figure maps onto itself as it symmetry.

rotates form 0° and 360° is called the order of symmetry. Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

The given figure has order of symmetry of 2, since The magnitude of symmetry is the smallest angle the figure can be rotated twice in 360°. through which a figure can be rotated so that it maps onto itself. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The figure has magnitude of symmetry of onto itself. .

Since the figure has order 2 rotational symmetry, the ANSWER: magnitude of the symmetry is . yes; 4; 90° ANSWER: yes; 2; 180°

State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

6. SOLUTION: A figure in the plane has rotational symmetry if the eSolutionsfigureManual can be- Powered mappedby Cogneroonto itself by a rotation Page 3 between 0° and 360° about the center of the figure. 7.

The given figure has rotational symmetry. SOLUTION: Vertical and horizontal lines through the center and diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way.

The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1.

Each quarter turn also maps the square onto itself. The number of times a figure maps onto itself as it So the rotations of 90, 180, and 270 degrees around rotates from 0° to 360° is called the order of the point (0, -1) map the square onto itself. symmetry. ANSWER: Since the figure can be rotated 4 times within 360° , line symmetry; rotational symmetry; the reflection in it has order 4 rotational symmetry the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in The magnitude of symmetry is the smallest angle the line y = -x - 1 map the square onto itself; the through which a figure can be rotated so that it maps rotations of 90, 180, and 270 degrees around the point onto itself. (0, -1) map the square onto itself.

The figure has magnitude of symmetry of .

ANSWER: yes; 4; 90°

8. SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent.

State whether the figure has line symmetry It does have rotational symmetry for each half turn and/or rotational symmetry. If so, describe the around its center, so a rotation of 180 degrees around reflections and/or rotations that map the figure the point (1, 1) maps the parallelogram onto itself. onto itself. ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

REGULARITY State whether the figure appears to have line symmetry. Write yes or no. 7. If so, copy the figure, draw all lines of SOLUTION: symmetry, and state their number. Vertical and horizontal lines through the center and diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way. 9. The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1. SOLUTION: A figure has reflectional symmetry if the figure can Each quarter turn also maps the square onto itself. be mapped onto itself by a reflection in a line. So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself. For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure ANSWER: does not have any lines of of symmetry. line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the ANSWER: reflection in the line y = x - 1, and the reflection in no the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

10. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

8. The given figure has reflectional symmetry. SOLUTION: This figure does not have line symmetry, because In order for the figure to map onto itself, the line of adjacent sides are not congruent. reflection must go through the center point.

It does have rotational symmetry for each half turn The figure has a vertical and horizontal line of around its center, so a rotation of 180 degrees around reflection. the point (1, 1) maps the parallelogram onto itself.

ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

REGULARITY State whether the figure It is also possible to have reflection over the appears to have line symmetry. Write yes or no. diagonal lines. If so, copy the figure, draw all lines of symmetry, and state their number.

9. SOLUTION: Therefore, the figure has four lines of symmetry A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry.

ANSWER: no ANSWER: yes; 4

10. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

In order for the figure to map onto itself, the line of

reflection must go through the center point. 11.

The figure has a vertical and horizontal line of SOLUTION: reflection. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point.

It is also possible to have reflection over the There are three lines of reflection that go though diagonal lines. opposites edges.

Therefore, the figure has four lines of symmetry There are three lines of reflection that go though opposites vertices.

ANSWER: There are six possible lines that go through the center yes; 4 and are lines of reflections. Thus, the hexagon has six lines of symmetry.

ANSWER: yes; 6 11. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point. 12. There are three lines of reflection that go though opposites edges. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

There is only one line of symmetry, a horizontal line through the middle of the figure. There are three lines of reflection that go though opposites vertices.

Thus, the figure has one line of symmetry.

ANSWER:

yes; 1 There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry.

13. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

ANSWER: The figure has reflectional symmetry. yes; 6 There is only one possible line of reflection, horizontally though the middle of the figure.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

12. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. 14.

There is only one line of symmetry, a horizontal line SOLUTION: through the middle of the figure. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. It is not possible to draw a line of reflection where the figure can map onto itself. Thus, the figure has one line of symmetry. ANSWER: ANSWER: no yes; 1 FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their number. 15. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can 13. be mapped onto itself by a reflection in a line. SOLUTION: A figure has reflectional symmetry if the figure can The flag does not have any reflectional symmetry. If be mapped onto itself by a reflection in a line. the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry. The figure has reflectional symmetry. ANSWER: There is only one possible line of reflection, no horizontally though the middle of the figure. 16. Refer to the flag on page 262. SOLUTION:

Thus, the figure has one line of symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. ANSWER: yes; 1 The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

14. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. It is not possible to draw a line of reflection where the figure can map onto itself.

ANSWER: Two diagonal lines of reflection are possible. no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their number. 15. Refer to page 262. SOLUTION:

A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line. There are a total of four possible lines that go through the center and are lines of reflections. Thus, The flag does not have any reflectional symmetry. If the flag has four lines of symmetry. the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER: no

16. Refer to the flag on page 262. SOLUTION: A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line. ANSWER:

yes; 4 The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

17. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

Two diagonal lines of reflection are possible. The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

It is not possible to reflect over a vertical or line

through the diagonals. There are a total of four possible lines that go through the center and are lines of reflections. Thus, Thus, the figure has one line of symmetry. the flag has four lines of symmetry. ANSWER: yes; 1

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state ANSWER: the order and magnitude of symmetry. yes; 4

18. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

17. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The figure has rotational symmetry.

The figure has reflectional symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. A horizontal line is a line of reflections for this flag.

This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto itself.

The magnitude of symmetry is the smallest angle It is not possible to reflect over a vertical or line through which a figure can be rotated so that it maps through the diagonals. onto itself.

Thus, the figure has one line of symmetry. The figure has a magnitude of symmetry of . ANSWER: ANSWER: yes; 1

yes; 2; 180°

A figure in the plane has rotational symmetry if the The triangle has rotational symmetry. figure can be mapped onto itself by a rotation

between 0° and 360° about the center of the figure.

The figure has rotational symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The number of times a figure maps onto itself as it symmetry. rotates from 0° to 360° is called the order of

symmetry.

The figure has order 3 rotational symmetry.

This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto The magnitude of symmetry is the smallest angle itself. through which a figure can be rotated so that it maps

onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The figure has magnitude of symmetry of onto itself. . ANSWER: The figure has a magnitude of symmetry of .

ANSWER:

yes; 3; 120°

yes; 2; 180°

20. SOLUTION: A figure in the plane has rotational symmetry if the 19. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. SOLUTION: A figure in the plane has rotational symmetry if the The isosceles trapezoid has no rotational symmetry. figure can be mapped onto itself by a rotation There is no way to rotate it such that it can be between 0° and 360° about the center of the figure. mapped onto itself.

The triangle has rotational symmetry. ANSWER: no

21.

The number of times a figure maps onto itself as it SOLUTION: rotates from 0° to 360° is called the order of A figure in the plane has rotational symmetry if the symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure has order 3 rotational symmetry. The crescent shaped figure has no rotational The magnitude of symmetry is the smallest angle symmetry. There is no way to rotate it such that it through which a figure can be rotated so that it maps can be mapped onto itself. onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 3; 120°

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself. ANSWER: no ANSWER: no

22. SOLUTION: 21. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The figure has rotational symmetry. between 0° and 360° about the center of the figure.

The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

ANSWER: no yes; 8; 45°

22. 23. SOLUTION: SOLUTION: A figure in the plane has rotational symmetry if the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. between 0° and 360° about the center of the figure.

The figure has rotational symmetry. The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of

The number of times a figure maps onto itself as it symmetry. rotates from 0° to 360° is called the order of symmetry. The figure has order 8 rotational symmetry. This means that the figure can be rotated 8 times and map

The figure has order 8 rotational symmetry. This onto itself within 360°. implies you can rotate the figure 8 times and have it map onto itself within 360°. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps

The magnitude of symmetry is the smallest angle onto itself. through which a figure can be rotated so that it maps onto itself. The figure has magnitude of symmetry of . The figure has magnitude of symmetry of ANSWER: .

ANSWER:

yes; 8; 45°

WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, yes; 8; 45° state the order and magnitude of symmetry. 24. Refer to page 263. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 23. between 0° and 360° about the center of the figure. SOLUTION:

The figure has rotational symmetry. symmetry.

The wheel has order 5 rotational symmetry. There are 5 large spokes and 5 small spokes. You can rotate the wheel 5 times within 360° and map the figure onto itself.

The magnitude of symmetry is the smallest angle The number of times a figure maps onto itself as it through which a figure can be rotated so that it maps rotates from 0° to 360° is called the order of onto itself. symmetry. The wheel has magnitude of symmetry The figure has order 8 rotational symmetry. This . means that the figure can be rotated 8 times and map onto itself within 360°. ANSWER: yes; 5; 72° The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps 25. Refer to page 263. onto itself. SOLUTION: The figure has magnitude of symmetry of A figure in the plane has rotational symmetry if the . figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. ANSWER: The wheel has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of

yes; 8; 45° symmetry.

WHEELS State whether each wheel cover appears The wheel has order 8 rotational symmetry. There to have rotational symmetry. Write yes or no. If so, are 8 spokes, thus the wheel can be rotated 8 times state the order and magnitude of symmetry. within 360° and map onto itself. 24. Refer to page 263. The magnitude of symmetry is the smallest angle SOLUTION: through which a figure can be rotated so that it maps A figure in the plane has rotational symmetry if the onto itself. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The wheel has order 8 rotational symmetry and magnitude . The wheel has rotational symmetry. ANSWER: The number of times a figure maps onto itself as it yes; 8; 45° rotates from 0° to 360° is called the order of symmetry. 26. Refer to page 263. SOLUTION: The wheel has order 5 rotational symmetry. There A figure in the plane has rotational symmetry if the are 5 large spokes and 5 small spokes. You can figure can be mapped onto itself by a rotation rotate the wheel 5 times within 360° and map the between 0° and 360° about the center of the figure. figure onto itself. The wheel has rotational symmetry.

symmetry. There are 10 bolts and the tire can be The wheel has magnitude of symmetry rotated 10 times within 360° and map onto itself. . ANSWER: The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps yes; 5; 72° onto itself. The wheel has magnitude of symmetry of . 25. Refer to page 263. ANSWER: SOLUTION: yes; 10; 36° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation State whether the figure has line symmetry between 0° and 360° about the center of the figure. and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure The wheel has rotational symmetry. onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself. 27.

SOLUTION: The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps This triangle is scalene, so it cannot have symmetry.

onto itself. ANSWER:

no symmetry The wheel has order 8 rotational symmetry and magnitude .

ANSWER: yes; 8; 45° 26. Refer to page 263.

SOLUTION: 28. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. This figure is a square, because each pair of adjacent The wheel has rotational symmetry. sides is congruent and perpendicular. All squares have both line and rotational symmetry. The number of times a figure maps onto itself as it The line symmetry is vertically, horizontally, and rotates from 0° to 360° is called the order of diagonally through the center of the square, with lines symmetry. The wheel has order 10 rotational that are either parallel to the sides of the square or symmetry. There are 10 bolts and the tire can be that include two vertices of the square. The rotated 10 times within 360° and map onto itself. equations of those lines are: x = 0, y = 0, y = x, and y = -x The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The rotational symmetry is for each quarter turn in a onto itself. The wheel has magnitude of symmetry of square, so the rotations of 90, 180, and 270 degrees . around the origin map the square onto itself.

ANSWER: ANSWER: yes; 10; 36° line symmetry; rotational symmetry; the reflection in State whether the figure has line symmetry the line x = 0, the reflection in the line y = 0, the and/or rotational symmetry. If so, describe the reflection in the line y = x, and the reflection in the reflections and/or rotations that map the figure line y = -x all map the square onto itself; the rotations onto itself. of 90, 180, and 270 degrees around the origin map the square onto itself.

27. 29. SOLUTION: SOLUTION: This triangle is scalene, so it cannot have symmetry. The trapezoid has line symmetry, because it is ANSWER: isosceles, but it does not have rotational symmetry, no symmetry because no trapezoid does.

The reflection in the line y = 1.5 maps the trapezoid onto itself, because that is the perpendicular bisector to the parallel sides.

ANSWER: line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself. 28. SOLUTION: This figure is a square, because each pair of adjacent sides is congruent and perpendicular. All squares have both line and rotational symmetry. The line symmetry is vertically, horizontally, and diagonally through the center of the square, with lines that are either parallel to the sides of the square or 30. that include two vertices of the square. The SOLUTION: equations of those lines are: x = 0, y = 0, y = x, and y = -x This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its The rotational symmetry is for each quarter turn in a center, which is the point (1, -1.5). square, so the rotations of 90, 180, and 270 degrees around the origin map the square onto itself. Since this parallelogram is not a rhombus it does not have line symmetry.

ANSWER: ANSWER: line symmetry; rotational symmetry; the reflection in rotational symmetry; the rotation of 180 degrees the line x = 0, the reflection in the line y = 0, the around the point (1, -1.5) maps the parallelogram reflection in the line y = x, and the reflection in the onto itself. line y = -x all map the square onto itself; the rotations of 90, 180, and 270 degrees around the origin map 31. MODELING Symmetry is an important component the square onto itself. of photography. Photographers often use reflection in water to create symmetry in photos. The photo on page 263 is a long exposure shot of the Eiffel tower reflected in a pool.

a. Describe the two-dimensional symmetry created by the photo. b. Is there rotational symmetry in the photo? Explain 29. your reasoning. SOLUTION: SOLUTION: The trapezoid has line symmetry, because it is a Sample answer: There is a horizontal line of isosceles, but it does not have rotational symmetry, symmetry between the tower and its reflection. because no trapezoid does. There is a vertical line of symmetry through the center of the photo. The reflection in the line y = 1.5 maps the trapezoid b No; sample answer: Because of how the image is onto itself, because that is the perpendicular bisector reflected over the horizontal line, there is no to the parallel sides. rotational symmetry.

ANSWER: ANSWER: line symmetry; the reflection in the line y = 1.5 maps a. Sample answer: There is a horizontal line of the trapezoid onto itself. symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no rotational symmetry.

COORDINATE GEOMETRY Determine 30. whether the figure with the given vertices has line symmetry and/or rotational symmetry. SOLUTION: 32. R(–3, 3), S(–3, –3), T(3, 3) This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its SOLUTION: center, which is the point (1, -1.5). Draw the figure on a coordinate plane.

Since this parallelogram is not a rhombus it does not have line symmetry.

ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram onto itself.

31. MODELING Symmetry is an important component A figure has line symmetry if the figure can be of photography. Photographers often use reflection in mapped onto itself by a reflection in a line. water to create symmetry in photos. The photo on page 263 is a long exposure shot of the Eiffel tower reflected in a pool. The given triangle has a line of symmetry through points (0, 0) and (–3, 3). a. Describe the two-dimensional symmetry created by the photo. A figure in the plane has rotational symmetry if the b. Is there rotational symmetry in the photo? Explain figure can be mapped onto itself by a rotation your reasoning. between 0° and 360° about the center of the figure. There is not way to rotate the figure and have it map SOLUTION: onto itself. a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. Thus, the figure has only line symmetry. There is a vertical line of symmetry through the center of the photo. ANSWER: b No; sample answer: Because of how the image is line reflected over the horizontal line, there is no rotational symmetry. 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) ANSWER: SOLUTION: a. Sample answer: There is a horizontal line of Draw the figure on a coordinate plane. symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no rotational symmetry.

COORDINATE GEOMETRY Determine

whether the figure with the given vertices has line symmetry and/or rotational symmetry. A figure has line symmetry if the figure can be 32. R(–3, 3), S(–3, –3), T(3, 3) mapped onto itself by a reflection in a line. The SOLUTION: given figure has 4 lines of symmetry. The line of Draw the figure on a coordinate plane. symmetry are though the following pairs of points {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, and {(2, 2), (2, –2)}.

Thus, the figure has both line symmetry and A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. rotational symmetry. ANSWER: The given triangle has a line of symmetry through line and rotational points (0, 0) and (–3, 3). 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – A figure in the plane has rotational symmetry if the 2) figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. There is not way to rotate the figure and have it map Draw the figure on a coordinate plane. onto itself.

Thus, the figure has only line symmetry.

ANSWER: line

33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) A figure has line symmetry if the figure can be SOLUTION: mapped onto itself by a reflection in a line. The Draw the figure on a coordinate plane. given hexagon has 2 lines of symmetry. The lines pass through the following pair of points {(0, 4), (0, – 4)}, and {(3, 0), (–3, 0)}

A figure has line symmetry if the figure can be Thus, the figure has both line symmetry and mapped onto itself by a reflection in a line. The rotational symmetry. given figure has 4 lines of symmetry. The line of symmetry are though the following pairs of points ANSWER: {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, line and rotational and {(2, 2), (2, –2)}. 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. Draw the figure on a coordinate plane. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational

34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – A figure has line symmetry if the figure can be 2) mapped onto itself by a reflection in a line. The SOLUTION: trapezoid has a line of reflection through points (0,3) and (0, –3). Draw the figure on a coordinate plane. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. There is no way to rotate this figure and have it map onto itself. Thus, it does not have rotational symmetry.

Therefore, the figure has only line symmetry. A figure has line symmetry if the figure can be ANSWER: mapped onto itself by a reflection in a line. The given hexagon has 2 lines of symmetry. The lines line pass through the following pair of points {(0, 4), (0, – ALGEBRA 4)}, and {(3, 0), (–3, 0)} Graph the function and determine whether the graph has line and/or rotational symmetry. If so, state the order and magnitude of A figure in the plane has rotational symmetry if the symmetry, and write the equations of any lines of figure can be mapped onto itself by a rotation symmetry. between 0° and 360° about the center of the figure.

The figure has rotational symmetry. You can rotate 36. y = x the figure once within 360° and have it map to itself. SOLUTION: Graph the function. Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) SOLUTION: Draw the figure on a coordinate plane.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation A figure has line symmetry if the figure can be between 0° and 360° about the center of the figure. mapped onto itself by a reflection in a line. The The line can be rotated twice within 360° and be trapezoid has a line of reflection through points (0,3) mapped onto itself. and (0, –3). The number of times a figure maps onto itself as it A figure in the plane has rotational symmetry if the rotates from 0° to 360° is called the order of figure can be mapped onto itself by a rotation symmetry. The graph has order 2 rotational between 0° and 360° about the center of the symmetry. figure. There is no way to rotate this figure and have it map onto itself. Thus, it does not have rotational The magnitude of symmetry is the smallest angle symmetry. through which a figure can be rotated so that it maps onto itself. Therefore, the figure has only line symmetry. The graph has magnitude of symmetry of . ANSWER: line Thus, the graph has both reflectional and rotational symmetry. ALGEBRA Graph the function and determine whether the graph has line and/or rotational ANSWER: symmetry. If so, state the order and magnitude of rotational; 2; 180°; line symmetry; y = –x symmetry, and write the equations of any lines of symmetry. 36. y = x SOLUTION: Graph the function.

2 37. y = x + 1 SOLUTION: Graph the function.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The line can be rotated twice within 360° and be A figure has reflectional symmetry if the figure can mapped onto itself. be mapped onto itself by a reflection in a line. The graph is reflected through the y-axis. Thus, the The number of times a figure maps onto itself as it equation of the line symmetry is x = 0. rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational A figure in the plane has rotational symmetry if the symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the The magnitude of symmetry is the smallest angle figure. There is no way to rotate the graph and have through which a figure can be rotated so that it maps it map onto itself. onto itself. The graph has magnitude of symmetry of Thus, the graph has only reflectional symmetry. . ANSWER: Thus, the graph has both reflectional and rotational line; x = 0 symmetry.

ANSWER: rotational; 2; 180°; line symmetry; y = –x

38. y = –x3 SOLUTION: Graph the function.

2 37. y = x + 1 SOLUTION: Graph the function.

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself.

A figure has reflectional symmetry if the figure can A figure in the plane has rotational symmetry if the be mapped onto itself by a reflection in a line. The figure can be mapped onto itself by a rotation graph is reflected through the y-axis. Thus, the between 0° and 360° about the center of the equation of the line symmetry is x = 0. figure. You can rotate the graph through the origin and have it map onto itself. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The number of times a figure maps onto itself as it between 0° and 360° about the center of the rotates from 0° to 360° is called the order of figure. There is no way to rotate the graph and have symmetry. The graph has order 2 rotational it map onto itself. symmetry.

Thus, the graph has only reflectional symmetry. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps ANSWER: onto itself. The graph has magnitude of symmetry of line; x = 0 .

Thus, the graph has only rotational symmetry.

ANSWER: rotational; 2; 180°

38. y = –x3 SOLUTION: Graph the function.

39. Refer to the rectangle on the coordinate plane.

A figure has reflectional symmetry if the figure can a. What are the equations of the lines of symmetry of be mapped onto itself by a reflection in a line. The the rectangle? graph does not have a line of reflections where the b. What happens to the equations of the lines of graph can be mapped onto itself. symmetry when the rectangle is rotated 90 degrees A figure in the plane has rotational symmetry if the counterclockwise around its center of symmetry? figure can be mapped onto itself by a rotation Explain. between 0° and 360° about the center of the SOLUTION: figure. You can rotate the graph through the origin and have it map onto itself. a. The lines of symmetry are parallel to the sides of the rectangles, and through the center of rotation. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The slopes of the sides of the rectangle are 0.5 and symmetry. The graph has order 2 rotational -2, so the slopes of the lines of symmetry are the symmetry. same. The center of the rectangle is (1, 1.5). Use the The magnitude of symmetry is the smallest angle point-slope formula to find equations. through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of

b. The equations of the lines of symmetry do not Thus, the graph has only rotational symmetry. change; although the rectangle does not map onto ANSWER: itself under this rotation, the lines of symmetry are rotational; 2; 180° mapped to each other. ANSWER: a. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to 39. Refer to the rectangle on the coordinate plane. investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. b. Geometric Use the rotation tool under the a. What are the equations of the lines of symmetry of transformation menu to investigate the rotational the rectangle? symmetry of the figure in part a. Then record its b. What happens to the equations of the lines of order of symmetry. symmetry when the rectangle is rotated 90 degrees c. Tabular Repeat the process in parts a and b for a counterclockwise around its center of symmetry? square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the Explain. order of symmetry for each polygon. SOLUTION: d. Verbal Make a conjecture about the number of a. The lines of symmetry are parallel to the sides of lines of symmetry and the order of symmetry for a the rectangles, and through the center of rotation. regular polygon with n sides. SOLUTION: The slopes of the sides of the rectangle are 0.5 and a. Construct an equilateral triangle and label the -2, so the slopes of the lines of symmetry are the vertices A, B, and C. Draw a line through A same. perpendicular to . Reflect the triangle in the line. The center of the rectangle is (1, 1.5). Use the Show the labels of the reflected image. If the image point-slope formula to find equations. maps to the original, then this line is a line of reflection.

b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

ANSWER: a. b. The equations of the lines of symmetry do not Next, draw a line through B perpendicular to . change; although the rectangle does not map onto Reflect the triangle in the line. Show the labels of the itself under this rotation, the lines of symmetry are reflected image. If the image maps to the original, mapped to each other. then this line is a line of reflection.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. Lastly, draw a line through C perpendicular to . b. Geometric Use the rotation tool under the Reflect the triangle in the line. Show the labels of the transformation menu to investigate the rotational reflected image. If the image maps to the original, symmetry of the figure in part a. Then record its then this line is a line of reflection. order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides. SOLUTION: a. Construct an equilateral triangle and label the vertices A, B, and C. Draw a line through A perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

There are 3 lines of symmetry.

Lastly, draw a line through C perpendicular to . Rotate the triangle again about point D. A 240 Reflect the triangle in the line. Show the labels of the degree rotation will map the image to the original. reflected image. If the image maps to the original, Show the labels of the image then this line is a line of reflection.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

There are 3 lines of symmetry.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Regular Pentagon

The triangle can be rotated a third time about D. A Construct a regular pentagon and then construct lines 360 degree rotation maps the image to the original. through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5.

Since the figure maps onto itself 3 times as it is rotated, the order of symmetry is 3. c. Square Construct a square and then construct lines through the midpoints of each side and diagonals. Use the

reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines Regular Hexagon constructed. So there are 4 lines of symmetry. Construct a regular hexagon and then construct lines Next, rotate the square about the center point. The through each vertex perpendicular to the sides. Use image maps to the original at 90, 180, 270, and 360 the reflection tool first to find that the image maps degree rotations. So the order of symmetry is 4. onto the original when reflected in each of the 6 lines constructed. So there are 6 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 60, 120, 180, 240, 300, and 360 degree rotations. So the order of symmetry is 6.

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5. d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and order of symmetry n.

ANSWER: a. 3 b. 3 c.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has Regular Hexagon only has line symmetry, and Jewel says that Figure A Construct a regular hexagon and then construct lines has only rotational symmetry. Is either of them through each vertex perpendicular to the sides. Use correct? Explain your reasoning. the reflection tool first to find that the image maps onto the original when reflected in each of the 6 lines constructed. So there are 6 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 60, 120, 180, 240, 300, and 360 degree rotations. So the order of symmetry

is 6. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. This figure has 4 lines of symmetry.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and

order of symmetry n. The figure also has rotational symmetry.

ANSWER: Therefore, neither of them are correct. Figure A has a. 3 both line and rotational symmetry. b. 3 ANSWER: c. Neither; Figure A has both line and rotational symmetry.

42. CHALLENGE A quadrilateral in the coordinate d. Sample answer: A regular polygon with n sides plane has exactly two lines of symmetry, y = x – 1 has n lines of symmetry and order of symmetry n. and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of 41. ERROR ANALYSIS Jaime says that Figure A has symmetry. only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them SOLUTION: correct? Explain your reasoning. Graph the figure and the lines of symmetry.

SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. This figure has 4 lines of symmetry. Pick points that are the same distance a from one line and the same distance b from the other line. In the same answer, the quadrilateral is a rectangle with sides which are parallel to the lines of symmetry. This guarantees that the vertices of the quadrilateral are the same distance a from one line and the same distance b from the other line. In this case, a =

A figure in the plane has rotational symmetry if the and b = . figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2). ANSWER: Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2)

The figure also has rotational symmetry.

Therefore, neither of them are correct. Figure A has both line and rotational symmetry.

ANSWER: Neither; Figure A has both line and rotational symmetry. 43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. 42. CHALLENGE A quadrilateral in the coordinate plane has exactly two lines of symmetry, y = x – 1 SOLUTION: and y = –x + 2. Find a set of possible vertices for circle; Every line through the center of a circle is a the figure. Graph the figure and the lines of line of symmetry, and there are infinitely many such symmetry. lines. SOLUTION: ANSWER: Graph the figure and the lines of symmetry. circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can Pick points that are the same distance a from one be mapped onto itself by a rotation between 0° and line and the same distance b from the other line. In 360° about the center of the figure. the same answer, the quadrilateral is a rectangle with sides which are parallel to the lines of symmetry. Identify a figure that has line symmetry but does not This guarantees that the vertices of the quadrilateral have rotational symmetry. are the same distance a from one line and the same distance b from the other line. In this case, a = An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does and b = . not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself. A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2).

ANSWER: Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) ANSWER: Sample answer: An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. SOLUTION: 45. WRITING IN MATH How are line symmetry and circle; Every line through the center of a circle is a rotational symmetry related? line of symmetry, and there are infinitely many such lines. SOLUTION: In both types of symmetries the figure is mapped ANSWER: onto itself. circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such Rotational symmetry. lines.

OPEN-ENDED 44. Draw a figure with line symmetry Reflectional symmetry: but not rotational symmetry. Explain. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and

360° about the center of the figure. In some cases an object can have both rotational and reflectional symmetry, such as the diamond, however Identify a figure that has line symmetry but does not some objects do not have both such as the crab. have rotational symmetry.

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

ANSWER: Sample answer: In both rotational and line symmetry

a figure is mapped onto itself. However, in line ANSWER: symmetry the figure is mapped onto itself by a Sample answer: An isosceles triangle has line reflection, and in rotational symmetry, a figure is symmetry from the vertex angle to the base of the mapped onto itself by a rotation. A figure can have triangle, but it does not have rotational symmetry line symmetry and rotational symmetry. because it cannot be rotated from 0° to 360° and 46. Sasha owns a tile store. For each tile in her store, she map onto itself. calculates the sum of the number of lines of symmetry and the order of symmetry, and then she enters this value into a database. Which value should she enter in the database for the tile shown here?

45. WRITING IN MATH How are line symmetry and rotational symmetry related?

SOLUTION: In both types of symmetries the figure is mapped onto itself. A 2 B 3 Rotational symmetry. C 4 D 8 SOLUTION: Reflectional symmetry:

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile. In some cases an object can have both rotational and reflectional symmetry, such as the diamond, however It has an order of symmetry of 2, because it has some objects do not have both such as the crab. rotational symmetry at 180 degrees, or each half turn around its center.

2 + 2 = 4, so C is the correct answer.

ANSWER: C

47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could ANSWER: be the figure that Patrick drew? Sample answer: In both rotational and line symmetry A a figure is mapped onto itself. However, in line symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have B line symmetry and rotational symmetry.

46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of C symmetry and the order of symmetry, and then she enters this value into a database. Which value should she enter in the database for the tile shown here? D

SOLUTION: A 2 B 3 Option A has rotational and reflectional symmetry. C 4 D 8 SOLUTION:

Option B has reflectional symmetry but not rotational symmetry.

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has Option C has neither rotational nor reflectional rotational symmetry at 180 degrees, or each half turn symmetry. around its center.

2 + 2 = 4, so C is the correct answer.

ANSWER: Option D has rotational symmetry but not reflectional C symmetry.

47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A Option E has reflectional symmetry but not rotational symmetry.

The correct choice is D. D ANSWER: D E 48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle SOLUTION: B Equiangular triangle C Isosceles triangle Option A has rotational and reflectional symmetry. D Scalene triangle SOLUTION: An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

Option B has reflectional symmetry but not rotational symmetry. ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS Option C has neither rotational nor reflectional has line symmetry but not rotational symmetry? symmetry. A B C D Option D has rotational symmetry but not reflectional symmetry. SOLUTION: First, plot the points.

Option E has reflectional symmetry but not rotational symmetry.

The correct choice is D. Then, plot each option A-D to consider each figure and its symmetry. ANSWER: Option A has both reflectional and rotational D symmetry.

ANSWER:

C Option B has reflective symmetry but not rotational symmetry. The correct choice is B. 49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D SOLUTION: First, plot the points.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

Thus, there are four possible lines that go through 1. the center and are lines of reflections.

SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point. Therefore, the figure has four lines of symmetry. Two lines of reflection go through the sides of the figure. ANSWER: yes; 4

Two lines of reflection go through the vertices of the figure.

2. SOLUTION: A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line.

Thus, there are four possible lines that go through The given figure does not have reflectional the center and are lines of reflections. symmetry. There is no way to fold or reflect it onto itself.

ANSWER: no

Therefore, the figure has four lines of symmetry. 3. SOLUTION: ANSWER: A figure has reflectional symmetry if the figure can yes; 4 be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

The figure has a vertical line of symmetry.

2. It does not have a horizontal line of symmetry. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto The figure does not have a line of symmetry through itself. the vertices. ANSWER: no

3. Thus, the figure has only one line of symmetry. SOLUTION: ANSWER: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. yes; 1

The given figure has reflectional symmetry.

The figure has a vertical line of symmetry.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

It does not have a horizontal line of symmetry.

4. SOLUTION:

A figure in the plane has rotational symmetry if the The figure does not have a line of symmetry through figure can be mapped onto itself by a rotation the vertices. between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: Thus, the figure has only one line of symmetry. no

ANSWER: yes; 1

5. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. State whether the figure has rotational symmetry. Write yes or no. If so, copy the The given figure has rotational symmetry. figure, locate the center of symmetry, and state

the order and magnitude of symmetry.

4. SOLUTION:

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The number of times a figure maps onto itself as it between 0° and 360° about the center of the figure. rotates form 0° and 360° is called the order of For the given figure, there is no rotation between 0° symmetry. and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational The given figure has order of symmetry of 2, since symmetry. the figure can be rotated twice in 360°.

ANSWER: The magnitude of symmetry is the smallest angle no through which a figure can be rotated so that it maps onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is .

5. ANSWER: SOLUTION: yes; 2; 180° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

6. SOLUTION: A figure in the plane has rotational symmetry if the

figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The number of times a figure maps onto itself as it

rotates form is called the order of 0° and 360° The given figure has rotational symmetry. symmetry.

The given figure has order of symmetry of 2, since the figure can be rotated twice in 360°.

The magnitude of symmetry is the smallest angle

through which a figure can be rotated so that it maps onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of Since the figure has order 2 rotational symmetry, the symmetry. magnitude of the symmetry is . Since the figure can be rotated 4 times within 360° , ANSWER: it has order 4 rotational symmetry yes; 2; 180° The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER: yes; 4; 90° 6. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry. State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. 7.

through which a figure can be rotated so that it maps The equations of those lines in this figure are x = 0, onto itself. y = -1, y = x - 1, and y = -x - 1.

The figure has magnitude of symmetry of Each quarter turn also maps the square onto itself. . So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself. ANSWER: yes; 4; 90° ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point 3-5 Symmetry (0, -1) map the square onto itself.

State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

8. SOLUTION: This figure does not have line symmetry, because 7. adjacent sides are not congruent.

SOLUTION: It does have rotational symmetry for each half turn Vertical and horizontal lines through the center and around its center, so a rotation of 180 degrees around diagonal lines through two opposite vertices are all the point (1, 1) maps the parallelogram onto itself. lines of symmetry for a square oriented this way. ANSWER: The equations of those lines in this figure are x = 0, rotational symmetry; the rotation of 180 degrees y = -1, y = x - 1, and y = -x - 1. around the point (1, 1) maps the parallelogram onto itself. Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around REGULARITY State whether the figure the point (0, -1) map the square onto itself. appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of ANSWER: symmetry, and state their number. line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point 9. (0, -1) map the square onto itself. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry.

ANSWER: 8. no SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around 10. the point (1, 1) maps the parallelogram onto itself. SOLUTION: ANSWER: A figure has reflectional line symmetry if the figure rotational symmetry; the rotation of 180 degrees can be mapped onto itself by a reflection in a line. around the point (1, 1) maps the parallelogram onto itself. The given figure has reflectional symmetry. eSolutions Manual - Powered by Cognero Page 4 REGULARITY State whether the figure In order for the figure to map onto itself, the line of appears to have line symmetry. Write yes or no. reflection must go through the center point. If so, copy the figure, draw all lines of symmetry, and state their number. The figure has a vertical and horizontal line of reflection.

9. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. It is also possible to have reflection over the For the given figure, there are no lines of reflection diagonal lines. where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry.

ANSWER: no

Therefore, the figure has four lines of symmetry

10. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry. ANSWER: In order for the figure to map onto itself, the line of yes; 4 reflection must go through the center point.

The figure has a vertical and horizontal line of reflection.

It is also possible to have reflection over the diagonal lines. 11. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry. Therefore, the figure has four lines of symmetry In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though opposites edges.

ANSWER: yes; 4

There are three lines of reflection that go though opposites vertices.

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has

11. six lines of symmetry. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point. ANSWER: There are three lines of reflection that go though yes; 6 opposites edges.

There are three lines of reflection that go though opposites vertices. 12. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has There is only one line of symmetry, a horizontal line six lines of symmetry. through the middle of the figure.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1 ANSWER: yes; 6

13. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

12. The figure has reflectional symmetry.

SOLUTION: There is only one possible line of reflection, A figure has reflectional symmetry if the figure can horizontally though the middle of the figure. be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

Thus, the figure has one line of symmetry. There is only one line of symmetry, a horizontal line through the middle of the figure. ANSWER: yes; 1

Thus, the figure has one line of symmetry.

ANSWER: yes; 1 14. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. It is not possible to draw a line of 13. reflection where the figure can map onto itself.

SOLUTION: ANSWER: A figure has reflectional symmetry if the figure can no be mapped onto itself by a reflection in a line. FLAGS State whether each flag design appears to The figure has reflectional symmetry. have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their There is only one possible line of reflection, number. horizontally though the middle of the figure. 15. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. Thus, the figure has one line of symmetry. The flag does not have any reflectional symmetry. If ANSWER: the red lines in the diagonals were in the same yes; 1 location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER:

16. Refer to the flag on page 262. 14. SOLUTION: SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The figure has reflectional symmetry.

The given figure does not have reflectional In order for the figure to map onto itself, the line of symmetry. It is not possible to draw a line of reflection must go through the center point. reflection where the figure can map onto itself.

ANSWER: A horizontal and vertical lines of reflection are no possible.

Two diagonal lines of reflection are possible. The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER: no

16. Refer to the flag on page 262.

SOLUTION: A figure has reflectional symmetry if the figure can There are a total of four possible lines that go be mapped onto itself by a reflection in a line. through the center and are lines of reflections. Thus, the flag has four lines of symmetry. The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

ANSWER: yes; 4

Two diagonal lines of reflection are possible.

17. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. There are a total of four possible lines that go through the center and are lines of reflections. Thus, The figure has reflectional symmetry. the flag has four lines of symmetry. A horizontal line is a line of reflections for this flag.

It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry. ANSWER: yes; 4 ANSWER: yes; 1

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state 17. Refer to page 262. the order and magnitude of symmetry. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. 18. SOLUTION: The figure has reflectional symmetry. A figure in the plane has rotational symmetry if the

figure can be mapped onto itself by a rotation

A horizontal line is a line of reflections for this flag. between 0° and 360° about the center of the figure.

It is not possible to reflect over a vertical or line through the diagonals. The figure has rotational symmetry.

Thus, the figure has one line of symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of ANSWER: symmetry. yes; 1 This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. State whether the figure has rotational symmetry. Write yes or no. If so, copy the The figure has a magnitude of symmetry of figure, locate the center of symmetry, and state . the order and magnitude of symmetry. ANSWER:

18. SOLUTION: A figure in the plane has rotational symmetry if the yes; 2; 180° figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

19. SOLUTION: A figure in the plane has rotational symmetry if the The figure has rotational symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The triangle has rotational symmetry. symmetry.

This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The number of times a figure maps onto itself as it The figure has a magnitude of symmetry of rotates from 0° to 360° is called the order of . symmetry.

ANSWER: The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

yes; 2; 180° The figure has magnitude of symmetry of .

ANSWER:

19. SOLUTION: A figure in the plane has rotational symmetry if the

figure can be mapped onto itself by a rotation yes; 3; 120° between 0° and 360° about the center of the figure.

The triangle has rotational symmetry.

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. The number of times a figure maps onto itself as it There is no way to rotate it such that it can be rotates from 0° to 360° is called the order of mapped onto itself. symmetry.

The figure has order 3 rotational symmetry. ANSWER: no The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of 21. . SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it yes; 3; 120° can be mapped onto itself.

20. SOLUTION: A figure in the plane has rotational symmetry if the

figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

ANSWER: no

21. SOLUTION: A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation no between 0° and 360° about the center of the figure.

The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself. 22. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps no onto itself.

The figure has magnitude of symmetry of .

ANSWER: 22. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry. yes; 8; 45°

23. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The number of times a figure maps onto itself as it between 0° and 360° about the center of the figure. rotates from 0° to 360° is called the order of

symmetry. The figure has rotational symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The figure has magnitude of symmetry of symmetry. . The figure has order 8 rotational symmetry. This ANSWER: means that the figure can be rotated 8 times and map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

yes; 8; 45° The figure has magnitude of symmetry of .

ANSWER:

23. SOLUTION:

A figure in the plane has rotational symmetry if the yes; 8; 45° figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, The figure has rotational symmetry. state the order and magnitude of symmetry. 24. Refer to page 263. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The number of times a figure maps onto itself as it The wheel has rotational symmetry. rotates from 0° to 360° is called the order of

symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The figure has order 8 rotational symmetry. This symmetry. means that the figure can be rotated 8 times and map

onto itself within 360°. The wheel has order 5 rotational symmetry. There are 5 large spokes and 5 small spokes. You can The magnitude of symmetry is the smallest angle rotate the wheel 5 times within 360° and map the through which a figure can be rotated so that it maps figure onto itself. onto itself. The magnitude of symmetry is the smallest angle The figure has magnitude of symmetry of through which a figure can be rotated so that it maps . onto itself.

ANSWER: The wheel has magnitude of symmetry .

ANSWER: yes; 5; 72° yes; 8; 45° 25. Refer to page 263. WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, SOLUTION: state the order and magnitude of symmetry. A figure in the plane has rotational symmetry if the 24. Refer to page 263. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. SOLUTION: A figure in the plane has rotational symmetry if the The wheel has rotational symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The wheel has rotational symmetry. symmetry.

The number of times a figure maps onto itself as it The wheel has order 8 rotational symmetry. There rotates from 0° to 360° is called the order of are 8 spokes, thus the wheel can be rotated 8 times symmetry. within 360° and map onto itself.

The wheel has order 5 rotational symmetry. There The magnitude of symmetry is the smallest angle are 5 large spokes and 5 small spokes. You can through which a figure can be rotated so that it maps rotate the wheel 5 times within 360° and map the onto itself. figure onto itself. The wheel has order 8 rotational symmetry and The magnitude of symmetry is the smallest angle magnitude . through which a figure can be rotated so that it maps onto itself. ANSWER: yes; 8; 45° The wheel has magnitude of symmetry . 26. Refer to page 263. ANSWER: SOLUTION: yes; 5; 72° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. 25. Refer to page 263. The wheel has rotational symmetry. SOLUTION: A figure in the plane has rotational symmetry if the The number of times a figure maps onto itself as it figure can be mapped onto itself by a rotation rotates from 0° to 360° is called the order of between 0° and 360° about the center of the figure. symmetry. The wheel has order 10 rotational symmetry. There are 10 bolts and the tire can be The wheel has rotational symmetry. rotated 10 times within 360° and map onto itself.

The number of times a figure maps onto itself as it The magnitude of symmetry is the smallest angle rotates from 0° to 360° is called the order of through which a figure can be rotated so that it maps symmetry. onto itself. The wheel has magnitude of symmetry of . The wheel has order 8 rotational symmetry. There ANSWER: are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself. yes; 10; 36° State whether the figure has line symmetry The magnitude of symmetry is the smallest angle and/or rotational symmetry. If so, describe the through which a figure can be rotated so that it maps reflections and/or rotations that map the figure onto itself. onto itself.

The wheel has order 8 rotational symmetry and magnitude .

ANSWER: yes; 8; 45° 26. Refer to page 263. 27. SOLUTION: SOLUTION: A figure in the plane has rotational symmetry if the This triangle is scalene, so it cannot have symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. ANSWER:

The wheel has rotational symmetry. no symmetry

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The wheel has order 10 rotational symmetry. There are 10 bolts and the tire can be rotated 10 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle

through which a figure can be rotated so that it maps 28. onto itself. The wheel has magnitude of symmetry of SOLUTION: . This figure is a square, because each pair of adjacent sides is congruent and perpendicular. ANSWER: All squares have both line and rotational symmetry. yes; 10; 36° The line symmetry is vertically, horizontally, and diagonally through the center of the square, with lines State whether the figure has line symmetry that are either parallel to the sides of the square or and/or rotational symmetry. If so, describe the that include two vertices of the square. The reflections and/or rotations that map the figure equations of those lines are: x = 0, y = 0, y = x, and y onto itself. = -x

The rotational symmetry is for each quarter turn in a square, so the rotations of 90, 180, and 270 degrees around the origin map the square onto itself.

ANSWER: 27. line symmetry; rotational symmetry; the reflection in SOLUTION: the line x = 0, the reflection in the line y = 0, the reflection in the line y = x, and the reflection in the This triangle is scalene, so it cannot have symmetry. line y = -x all map the square onto itself; the rotations ANSWER: of 90, 180, and 270 degrees around the origin map the square onto itself. no symmetry

29. 28. SOLUTION: SOLUTION: The trapezoid has line symmetry, because it is This figure is a square, because each pair of adjacent isosceles, but it does not have rotational symmetry, sides is congruent and perpendicular. because no trapezoid does. All squares have both line and rotational symmetry. The line symmetry is vertically, horizontally, and The reflection in the line y = 1.5 maps the trapezoid diagonally through the center of the square, with lines onto itself, because that is the perpendicular bisector that are either parallel to the sides of the square or to the parallel sides. that include two vertices of the square. The equations of those lines are: x = 0, y = 0, y = x, and y ANSWER: = -x line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself. The rotational symmetry is for each quarter turn in a square, so the rotations of 90, 180, and 270 degrees around the origin map the square onto itself.

ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = 0, the

reflection in the line y = x, and the reflection in the 30. line y = -x all map the square onto itself; the rotations SOLUTION: of 90, 180, and 270 degrees around the origin map This figure is a parallelogram, so it has rotational the square onto itself. symmetry of a half turn or 180 degrees around its center, which is the point (1, -1.5).

Since this parallelogram is not a rhombus it does not have line symmetry.

ANSWER: rotational symmetry; the rotation of 180 degrees 29. around the point (1, -1.5) maps the parallelogram SOLUTION: onto itself. The trapezoid has line symmetry, because it is 31. MODELING Symmetry is an important component isosceles, but it does not have rotational symmetry, of photography. Photographers often use reflection in because no trapezoid does. water to create symmetry in photos. The photo on page 263 is a long exposure shot of the Eiffel tower The reflection in the line y = 1.5 maps the trapezoid reflected in a pool. onto itself, because that is the perpendicular bisector to the parallel sides. a. Describe the two-dimensional symmetry created ANSWER: by the photo. line symmetry; the reflection in the line y = 1.5 maps b. Is there rotational symmetry in the photo? Explain the trapezoid onto itself. your reasoning. SOLUTION: a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no

30. rotational symmetry. SOLUTION: ANSWER: This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its a. Sample answer: There is a horizontal line of center, which is the point (1, -1.5). symmetry between the tower and its reflection. There is a vertical line of symmetry through the Since this parallelogram is not a rhombus it does not center of the photo. have line symmetry. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no ANSWER: rotational symmetry. rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram COORDINATE GEOMETRY Determine onto itself. whether the figure with the given vertices has line symmetry and/or rotational symmetry. 31. MODELING Symmetry is an important component 32. R(–3, 3), S(–3, –3), T(3, 3) of photography. Photographers often use reflection in SOLUTION: water to create symmetry in photos. The photo on page 263 is a long exposure shot of the Eiffel tower Draw the figure on a coordinate plane. reflected in a pool.

a. Describe the two-dimensional symmetry created by the photo. b. Is there rotational symmetry in the photo? Explain your reasoning. SOLUTION: a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. A figure has line symmetry if the figure can be There is a vertical line of symmetry through the mapped onto itself by a reflection in a line. center of the photo. b No; sample answer: Because of how the image is The given triangle has a line of symmetry through reflected over the horizontal line, there is no points (0, 0) and (–3, 3). rotational symmetry. A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation a. Sample answer: There is a horizontal line of between 0° and 360° about the center of the figure. symmetry between the tower and its reflection. There is not way to rotate the figure and have it map There is a vertical line of symmetry through the onto itself. center of the photo. b No; sample answer: Because of how the image is Thus, the figure has only line symmetry. reflected over the horizontal line, there is no rotational symmetry. ANSWER: line COORDINATE GEOMETRY Determine whether the figure with the given vertices has line symmetry and/or rotational symmetry. 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) 32. R(–3, 3), S(–3, –3), T(3, 3) SOLUTION: SOLUTION: Draw the figure on a coordinate plane. Draw the figure on a coordinate plane.

A figure has line symmetry if the figure can be A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The mapped onto itself by a reflection in a line. given figure has 4 lines of symmetry. The line of symmetry are though the following pairs of points The given triangle has a line of symmetry through {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, and {(2, 2), (2, –2)}. points (0, 0) and (–3, 3).

A figure in the plane has rotational symmetry if the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map There is not way to rotate the figure and have it map onto itself. The order of symmetry is 4.

onto itself.

Thus, the figure has both line symmetry and Thus, the figure has only symmetry. line rotational symmetry. ANSWER: ANSWER: line line and rotational 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 2) SOLUTION: Draw the figure on a coordinate plane. SOLUTION: Draw the figure on a coordinate plane.

A figure has line symmetry if the figure can be A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The mapped onto itself by a reflection in a line. The given figure has 4 lines of symmetry. The line of given hexagon has 2 lines of symmetry. The lines symmetry are though the following pairs of points pass through the following pair of points {(0, 4), (0, – {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, 4)}, and {(3, 0), (–3, 0)} and {(2, 2), (2, –2)}. A figure in the plane has rotational symmetry if the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. between 0° and 360° about the center of the figure. The figure has rotational symmetry. You can rotate The figure can be rotated from the origin and map the figure once within 360° and have it map to itself. onto itself. The order of symmetry is 4.

Thus, the figure has both line symmetry and Thus, the figure has both line symmetry and rotational symmetry. rotational symmetry. ANSWER: ANSWER: line and rotational line and rotational 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 2) SOLUTION: Draw the figure on a coordinate plane. SOLUTION: Draw the figure on a coordinate plane.

A figure has line symmetry if the figure can be A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The mapped onto itself by a reflection in a line. The trapezoid has a line of reflection through points (0,3) given hexagon has 2 lines of symmetry. The lines and (0, –3). pass through the following pair of points {(0, 4), (0, – 4)}, and {(3, 0), (–3, 0)} A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation A figure in the plane has rotational symmetry if the between 0° and 360° about the center of the figure can be mapped onto itself by a rotation figure. There is no way to rotate this figure and have between 0° and 360° about the center of the figure. it map onto itself. Thus, it does not have rotational The figure has rotational symmetry. You can rotate symmetry. the figure once within 360° and have it map to itself. Therefore, the figure has only line symmetry. Thus, the figure has both line symmetry and rotational symmetry. ANSWER: line ANSWER: line and rotational ALGEBRA Graph the function and determine whether the graph has line and/or rotational 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) symmetry. If so, state the order and magnitude of symmetry, and write the equations of any lines of SOLUTION: symmetry. Draw the figure on a coordinate plane. 36. y = x SOLUTION: Graph the function.

A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The trapezoid has a line of reflection through points (0,3) and (0, –3). A figure has reflectional symmetry if the figure can A figure in the plane has rotational symmetry if the be mapped onto itself by a reflection in a line. The figure can be mapped onto itself by a rotation line y = x has reflectional symmetry since any line between 0° and 360° about the center of the perpendicular to y = x is a line of reflection. The figure. There is no way to rotate this figure and have equation of the line symmetry is y = –x. it map onto itself. Thus, it does not have rotational symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation Therefore, the figure has only line symmetry. between 0° and 360° about the center of the figure. The line can be rotated twice within 360° and be ANSWER: mapped onto itself. line The number of times a figure maps onto itself as it ALGEBRA Graph the function and determine rotates from 0° to 360° is called the order of whether the graph has line and/or rotational symmetry. The graph has order 2 rotational symmetry. If so, state the order and magnitude of symmetry. symmetry, and write the equations of any lines of symmetry. The magnitude of symmetry is the smallest angle 36. y = x through which a figure can be rotated so that it maps SOLUTION: onto itself. Graph the function. The graph has magnitude of symmetry of .

Thus, the graph has both reflectional and rotational symmetry.

ANSWER: rotational; 2; 180°; line symmetry; y = –x

A figure in the plane has rotational symmetry if the 2 figure can be mapped onto itself by a rotation 37. y = x + 1 between 0° and 360° about the center of the figure. The line can be rotated twice within 360° and be SOLUTION: mapped onto itself. Graph the function.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry.

ANSWER: A figure in the plane has rotational symmetry if the rotational; 2; 180°; line symmetry; y = –x figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. There is no way to rotate the graph and have it map onto itself.

Thus, the graph has only reflectional symmetry.

ANSWER: line; x = 0

2 37. y = x + 1 SOLUTION: Graph the function.

38. y = –x3 SOLUTION: Graph the function.

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph is reflected through the y-axis. Thus, the equation of the line symmetry is x = 0.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps 3 38. y = –x onto itself. The graph has magnitude of symmetry of SOLUTION: . Graph the function. Thus, the graph has only rotational symmetry.

ANSWER: rotational; 2; 180°

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself. 39. Refer to the rectangle on the coordinate plane. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. You can rotate the graph through the origin and have it map onto itself.

a. The lines of symmetry are parallel to the sides of Thus, the graph has only rotational symmetry. the rectangles, and through the center of rotation. ANSWER: rotational; 2; 180° The slopes of the sides of the rectangle are 0.5 and -2, so the slopes of the lines of symmetry are the same. The center of the rectangle is (1, 1.5). Use the point-slope formula to find equations.

b. The equations of the lines of symmetry do not

change; although the rectangle does not map onto 39. Refer to the rectangle on the coordinate plane. itself under this rotation, the lines of symmetry are mapped to each other.

ANSWER: a. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other. a. What are the equations of the lines of symmetry of the rectangle? 40. MULTIPLE REPRESENTATIONS In this b. What happens to the equations of the lines of problem, you will use dynamic geometric software to symmetry when the rectangle is rotated 90 degrees investigate line and rotational symmetry in regular counterclockwise around its center of symmetry? polygons. Explain. a. Geometric Use The Geometer’s Sketchpad to SOLUTION: draw an equilateral triangle. Use the reflection tool a. The lines of symmetry are parallel to the sides of under the transformation menu to investigate and the rectangles, and through the center of rotation. determine all possible lines of symmetry. Then record their number. The slopes of the sides of the rectangle are 0.5 and b. Geometric Use the rotation tool under the -2, so the slopes of the lines of symmetry are the transformation menu to investigate the rotational a same. symmetry of the figure in part . Then record its order of symmetry. The center of the rectangle is (1, 1.5). Use the c. Tabular Repeat the process in parts a and b for a point-slope formula to find equations. square, regular pentagon, and regular hexagon.

Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of b. The equations of the lines of symmetry do not lines of symmetry and the order of symmetry for a regular polygon with n sides. change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are SOLUTION: mapped to each other. a. Construct an equilateral triangle and label the vertices A, B, and C. Draw a line through A ANSWER: perpendicular to . Reflect the triangle in the line. a. Show the labels of the reflected image. If the image b. The equations of the lines of symmetry do not maps to the original, then this line is a line of change; although the rectangle does not map onto reflection. itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record Next, draw a line through B perpendicular to . their number. Reflect the triangle in the line. Show the labels of the b. Geometric Use the rotation tool under the reflected image. If the image maps to the original, transformation menu to investigate the rotational then this line is a line of reflection. symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides. Lastly, draw a line through C perpendicular to . SOLUTION: Reflect the triangle in the line. Show the labels of the a. Construct an equilateral triangle and label the reflected image. If the image maps to the original, vertices A, B, and C. Draw a line through A then this line is a line of reflection. perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the

reflected image. If the image maps to the original, There are 3 lines of symmetry. then this line is a line of reflection. b. Construct an equilateral triangle and show the labels of the vertices. Next, find the center of the triangle. Since this is an equilateral triangle, the circumcenter, incenter, centroid, and orthocenter are the same point. Construct altitudes through each vertex and label the intersection. Rotate the triangle about point D. A 120 degree rotation will map the image to the original. Show the labels of the image. Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

There are 3 lines of symmetry.

Rotate the triangle again about point D. A 240 Since the figure maps onto itself 3 times as it is degree rotation will map the image to the original. rotated, the order of symmetry is 3. Show the labels of the image c. Square Construct a square and then construct lines through the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

Since the figure maps onto itself 3 times as it is constructed. So there are 5 lines of symmetry. rotated, the order of symmetry is 3. Next, rotate the square about the center point. The c. image maps to the original at 72, 144, 216, 288, and Square 360 degree rotations. So the order of symmetry is 5. Construct a square and then construct lines through the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Next, rotate the square about the center point. The Regular Pentagon image maps to the original at 60, 120, 180, 240, 300, Construct a regular pentagon and then construct lines and 360 degree rotations. So the order of symmetry through each vertex perpendicular to the sides. Use is 6. the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5.

d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and order of symmetry n. Regular Hexagon Construct a regular hexagon and then construct lines ANSWER: through each vertex perpendicular to the sides. Use a. 3 the reflection tool first to find that the image maps b. 3 onto the original when reflected in each of the 6 lines c. constructed. So there are 6 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 60, 120, 180, 240, 300, and 360 degree rotations. So the order of symmetry is 6. d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

SOLUTION:

A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. This figure has 4 lines of symmetry.

ANSWER: a. 3 A figure in the plane has rotational symmetry if the b. 3 figure can be mapped onto itself by a rotation c. between 0° and 360° about the center of the figure.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

The figure also has rotational symmetry. ERROR ANALYSIS 41. Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them Therefore, neither of them are correct. Figure A has correct? Explain your reasoning. both line and rotational symmetry. ANSWER: Neither; Figure A has both line and rotational symmetry.

42. CHALLENGE A quadrilateral in the coordinate

plane has exactly two lines of symmetry, y = x – 1 SOLUTION: and y = –x + 2. Find a set of possible vertices for A figure has line symmetry if the figure can be the figure. Graph the figure and the lines of mapped onto itself by a reflection in a line. This symmetry. figure has 4 lines of symmetry. SOLUTION: Graph the figure and the lines of symmetry.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Pick points that are the same distance a from one line and the same distance b from the other line. In the same answer, the quadrilateral is a rectangle with sides which are parallel to the lines of symmetry. This guarantees that the vertices of the quadrilateral are the same distance a from one line and the same The figure also has rotational symmetry. distance b from the other line. In this case, a = Therefore, neither of them are correct. Figure A has and b = . both line and rotational symmetry.

ANSWER: A set of possible vertices for the figure are, (–1, 0), Neither; Figure A has both line and rotational (2, 3), (4, 1), and (1, 2). symmetry. ANSWER: 42. CHALLENGE A quadrilateral in the coordinate Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) plane has exactly two lines of symmetry, y = x – 1 and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of symmetry. SOLUTION: Graph the figure and the lines of symmetry.

43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. SOLUTION: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines. Pick points that are the same distance a from one line and the same distance b from the other line. In ANSWER: the same answer, the quadrilateral is a rectangle with circle; Every line through the center of a circle is a sides which are parallel to the lines of symmetry. line of symmetry, and there are infinitely many such This guarantees that the vertices of the quadrilateral lines. are the same distance a from one line and the same 44. OPEN-ENDED Draw a figure with line symmetry distance b from the other line. In this case, a = but not rotational symmetry. Explain. and b = . SOLUTION: A figure has line symmetry if the figure can be A set of possible vertices for the figure are, (–1, 0), mapped onto itself by a reflection in a line. A figure (2, 3), (4, 1), and (1, 2). in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and ANSWER: 360° about the center of the figure. Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) Identify a figure that has line symmetry but does not have rotational symmetry.

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

43. REASONING A figure has infinitely many lines of

symmetry. What is the figure? Explain. SOLUTION: ANSWER: circle; Every line through the center of a circle is a Sample answer: An isosceles triangle has line line of symmetry, and there are infinitely many such symmetry from the vertex angle to the base of the lines. triangle, but it does not have rotational symmetry ANSWER: because it cannot be rotated from 0° to 360° and map onto itself. circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. SOLUTION: 45. WRITING IN MATH How are line symmetry and A figure has line symmetry if the figure can be rotational symmetry related? mapped onto itself by a reflection in a line. A figure SOLUTION: in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and In both types of symmetries the figure is mapped 360° about the center of the figure. onto itself.

Identify a figure that has line symmetry but does not Rotational symmetry. have rotational symmetry.

An isosceles triangle has line symmetry from the Reflectional symmetry: vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

In some cases an object can have both rotational and reflectional symmetry, such as the diamond, however ANSWER: some objects do not have both such as the crab.

Sample answer: An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

ANSWER: Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line 45. WRITING IN MATH How are line symmetry and symmetry the figure is mapped onto itself by a rotational symmetry related? reflection, and in rotational symmetry, a figure is SOLUTION: mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry. In both types of symmetries the figure is mapped onto itself. 46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of Rotational symmetry. symmetry and the order of symmetry, and then she enters this value into a database. Which value should she enter in the database for the tile shown here?

Reflectional symmetry:

A 2 In some cases an object can have both rotational and B 3 reflectional symmetry, such as the diamond, however C 4 some objects do not have both such as the crab. D 8 SOLUTION:

The tile is a rhombus and has 2 lines of symmetry. ANSWER: Each connects opposite corners of the tile. Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line It has an order of symmetry of 2, because it has symmetry the figure is mapped onto itself by a rotational symmetry at 180 degrees, or each half turn reflection, and in rotational symmetry, a figure is around its center. mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry. 2 + 2 = 4, so C is the correct answer.

46. Sasha owns a tile store. For each tile in her store, she ANSWER: calculates the sum of the number of lines of C symmetry and the order of symmetry, and then she enters this value into a database. Which value should 47. Patrick drew a figure that has rotational symmetry she enter in the database for the tile shown here? but not line symmetry. Which of the following could be the figure that Patrick drew? A

A 2

B 3 C C 4 D 8 SOLUTION: D

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile. SOLUTION: Option A has rotational and reflectional symmetry. It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn around its center.

2 + 2 = 4, so C is the correct answer. Option B has reflectional symmetry but not rotational ANSWER: symmetry. C

47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A Option C has neither rotational nor reflectional symmetry.

Option D has rotational symmetry but not reflectional symmetry. C

Option E has reflectional symmetry but not rotational E symmetry.

SOLUTION: Option A has rotational and reflectional symmetry.

The correct choice is D.

ANSWER: Option B has reflectional symmetry but not rotational D symmetry. 48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle Option C has neither rotational nor reflectional D Scalene triangle symmetry. SOLUTION: An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

Option D has rotational symmetry but not reflectional symmetry. ANSWER: C

symmetry. A B C D SOLUTION: First, plot the points.

The correct choice is D.

ANSWER: D

48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle D Scalene triangle SOLUTION: Then, plot each option A-D to consider each figure An isosceles triangle has one line of symmetry and and its symmetry. no rotational symmetry. The correct choice is C. Option A has both reflectional and rotational

symmetry. ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D

SOLUTION: First, plot the points. Option B has reflective symmetry but not rotational symmetry. The correct choice is B.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

Thus, there are four possible lines that go through the center and are lines of reflections.

Therefore, the figure has four lines of symmetry.

State whether the figure appears to have line ANSWER: symmetry. Write yes or no. If so, copy the yes; 4 figure, draw all lines of symmetry, and state their number.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. 2.

In order for the figure to map onto itself, the line of SOLUTION: reflection must go through the center point. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. Two lines of reflection go through the sides of the figure. The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER: no

Two lines of reflection go through the vertices of the figure. 3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

The figure has a vertical line of symmetry. Thus, there are four possible lines that go through the center and are lines of reflections.

It does not have a horizontal line of symmetry.

Therefore, the figure has four lines of symmetry.

ANSWER: yes; 4 The figure does not have a line of symmetry through the vertices.

Thus, the figure has only one line of symmetry.

ANSWER: 2. yes; 1 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

3. 4. SOLUTION: SOLUTION: A figure has reflectional symmetry if the figure can A figure in the plane has rotational symmetry if the be mapped onto itself by a reflection in a line. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The given figure has reflectional symmetry. For the given figure, there is no rotation between 0° The figure has a vertical line of symmetry. and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: no

It does not have a horizontal line of symmetry.

SOLUTION: A figure in the plane has rotational symmetry if the The figure does not have a line of symmetry through figure can be mapped onto itself by a rotation the vertices. between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

Thus, the figure has only one line of symmetry.

ANSWER: yes; 1 The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of symmetry.

The given figure has order of symmetry of 2, since the figure can be rotated twice in 360°.

ANSWER:

4. yes; 2; 180° SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: 6. no SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The number of times a figure maps onto itself as it onto itself.

rotates form 0° and 360° is called the order of The figure has magnitude of symmetry of symmetry. .

The given figure has order of symmetry of 2, since ANSWER: the figure can be rotated twice in 360°. yes; 4; 90°

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is .

ANSWER: State whether the figure has line symmetry and/or rotational symmetry. If so, describe the yes; 2; 180° reflections and/or rotations that map the figure onto itself.

7. SOLUTION:

6. Vertical and horizontal lines through the center and SOLUTION: diagonal lines through two opposite vertices are all A figure in the plane has rotational symmetry if the lines of symmetry for a square oriented this way. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1.

The given figure has rotational symmetry. Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of 8. . SOLUTION: This figure does not have line symmetry, because ANSWER: adjacent sides are not congruent. yes; 4; 90° It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

9. 7. SOLUTION: SOLUTION: A figure has reflectional symmetry if the figure can

Vertical and horizontal lines through the center and be mapped onto itself by a reflection in a line. diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way. For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure The equations of those lines in this figure are x = 0, does not have any lines of of symmetry. y = -1, y = x - 1, and y = -x - 1. ANSWER:

Each quarter turn also maps the square onto itself. no So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

The given figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

The figure has a vertical and horizontal line of reflection. 8. SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent.

REGULARITY State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of Therefore, the figure has four lines of symmetry symmetry, and state their number.

9. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. ANSWER:

yes; 4 For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry.

ANSWER: no

10. SOLUTION: 11. 3-5 SymmetryA figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line. SOLUTION: A figure has reflectional symmetry if the figure can The given figure has reflectional symmetry. be mapped onto itself by a reflection in a line.

It is also possible to have reflection over the diagonal lines. There are three lines of reflection that go though opposites vertices.

Therefore, the figure has four lines of symmetry

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry.

ANSWER: yes; 4

ANSWER: yes; 6

11. SOLUTION: A figure has reflectional symmetry if the figure can 12. eSolutionsbe mapped onto itself by a reflection in a line. Manual - Powered by Cognero Page 5 SOLUTION: The given hexagon has reflectional symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. In order for the hexagon to map onto itself, the line of reflection must go through the center point. The figure has reflectional symmetry.

There are three lines of reflection that go though There is only one line of symmetry, a horizontal line opposites edges. through the middle of the figure.

Thus, the figure has one line of symmetry.

There are three lines of reflection that go though ANSWER: opposites vertices. yes; 1

There are six possible lines that go through the center 13. and are lines of reflections. Thus, the hexagon has SOLUTION: six lines of symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

There is only one possible line of reflection, horizontally though the middle of the figure.

ANSWER: yes; 6 Thus, the figure has one line of symmetry. ANSWER: yes; 1

14. 12. SOLUTION: SOLUTION: A figure has reflectional symmetry if the figure can A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. be mapped onto itself by a reflection in a line. The given figure does not have reflectional The figure has reflectional symmetry. symmetry. It is not possible to draw a line of reflection where the figure can map onto itself. There is only one line of symmetry, a horizontal line ANSWER: through the middle of the figure. no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their

Thus, the figure has one line of symmetry. number. 15. Refer to page 262. ANSWER: SOLUTION: yes; 1 A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

13. ANSWER: no SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. 16. Refer to the flag on page 262. SOLUTION: The figure has reflectional symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. There is only one possible line of reflection, horizontally though the middle of the figure. The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of

reflection must go through the center point. Thus, the figure has one line of symmetry. A horizontal and vertical lines of reflection are ANSWER: possible. yes; 1

14. SOLUTION: Two diagonal lines of reflection are possible. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. It is not possible to draw a line of reflection where the figure can map onto itself.

ANSWER: no

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER: ANSWER: no yes; 4

16. Refer to the flag on page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

Two diagonal lines of reflection are possible.

It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

There are a total of four possible lines that go through the center and are lines of reflections. Thus, the flag has four lines of symmetry.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

18. SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the yes; 4 figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

itself.

The figure has a magnitude of symmetry of .

ANSWER:

It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry. yes; 2; 180° ANSWER: yes; 1

19. SOLUTION:

A figure in the plane has rotational symmetry if the State whether the figure has rotational figure can be mapped onto itself by a rotation symmetry. Write yes or no. If so, copy the between 0° and 360° about the center of the figure. figure, locate the center of symmetry, and state the order and magnitude of symmetry. The triangle has rotational symmetry.

The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle The figure has rotational symmetry. through which a figure can be rotated so that it maps

onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The figure has magnitude of symmetry of symmetry. . This figure has order 2 rotational symmetry, since ANSWER: you have to rotate 180° to get the figure to map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. yes; 3; 120°

The figure has a magnitude of symmetry of .

19. ANSWER: SOLUTION: no A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The triangle has rotational symmetry.

21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it The number of times a figure maps onto itself as it can be mapped onto itself. rotates from 0° to 360° is called the order of symmetry.

The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 3; 120°

20. SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the no figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The crescent shaped figure has no rotational The number of times a figure maps onto itself as it symmetry. There is no way to rotate it such that it rotates from 0° to 360° is called the order of can be mapped onto itself. symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 8; 45°

23. ANSWER: no SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This means that the figure can be rotated 8 times and map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The number of times a figure maps onto itself as it onto itself. rotates from 0° to 360° is called the order of symmetry. The figure has magnitude of symmetry of The figure has order 8 rotational symmetry. This . implies you can rotate the figure 8 times and have it ANSWER: map onto itself within 360°.

yes; 8; 45° The wheel has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. 23. SOLUTION: The wheel has order 5 rotational symmetry. There are 5 large spokes and 5 small spokes. You can A figure in the plane has rotational symmetry if the rotate the wheel 5 times within 360° and map the figure can be mapped onto itself by a rotation figure onto itself. between 0° and 360° about the center of the figure.

The magnitude of symmetry is the smallest angle The figure has rotational symmetry. through which a figure can be rotated so that it maps onto itself.

The wheel has magnitude of symmetry .

ANSWER: The number of times a figure maps onto itself as it yes; 5; 72° rotates from 0° to 360° is called the order of symmetry. 25. Refer to page 263.

ANSWER: The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps yes; 8; 45° onto itself. WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, The wheel has order 8 rotational symmetry and state the order and magnitude of symmetry. magnitude . 24. Refer to page 263. ANSWER: SOLUTION: yes; 8; 45° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 26. Refer to page 263.

between 0° and 360° about the center of the figure. SOLUTION:

A figure in the plane has rotational symmetry if the The wheel has rotational symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The number of times a figure maps onto itself as it The wheel has rotational symmetry. rotates from 0° to 360° is called the order of symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The wheel has order 5 rotational symmetry. There symmetry. The wheel has order 10 rotational are 5 large spokes and 5 small spokes. You can symmetry. There are 10 bolts and the tire can be rotate the wheel 5 times within 360° and map the rotated 10 times within 360° and map onto itself. figure onto itself. The magnitude of symmetry is the smallest angle The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps through which a figure can be rotated so that it maps onto itself. The wheel has magnitude of symmetry of onto itself. .

SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The wheel has rotational symmetry. 27. The number of times a figure maps onto itself as it SOLUTION: rotates from 0° to 360° is called the order of symmetry. This triangle is scalene, so it cannot have symmetry. ANSWER: The wheel has order 8 rotational symmetry. There no symmetry are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The wheel has order 8 rotational symmetry and 28. magnitude . SOLUTION: ANSWER: This figure is a square, because each pair of adjacent yes; 8; 45° sides is congruent and perpendicular. All squares have both line and rotational symmetry. 26. Refer to page 263. The line symmetry is vertically, horizontally, and SOLUTION: diagonally through the center of the square, with lines A figure in the plane has rotational symmetry if the that are either parallel to the sides of the square or figure can be mapped onto itself by a rotation that include two vertices of the square. The between 0° and 360° about the center of the figure. equations of those lines are: x = 0, y = 0, y = x, and y The wheel has rotational symmetry. = -x

line symmetry; rotational symmetry; the reflection in The magnitude of symmetry is the smallest angle the line x = 0, the reflection in the line y = 0, the through which a figure can be rotated so that it maps reflection in the line y = x, and the reflection in the onto itself. The wheel has magnitude of symmetry of line y = -x all map the square onto itself; the rotations . of 90, 180, and 270 degrees around the origin map the square onto itself. ANSWER: yes; 10; 36° State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

29. SOLUTION: The trapezoid has line symmetry, because it is isosceles, but it does not have rotational symmetry, because no trapezoid does.

27. The reflection in the line y = 1.5 maps the trapezoid SOLUTION: onto itself, because that is the perpendicular bisector to the parallel sides. This triangle is scalene, so it cannot have symmetry. ANSWER: ANSWER: line symmetry; the reflection in the line y = 1.5 maps no symmetry the trapezoid onto itself.

28. 30. SOLUTION: SOLUTION: This figure is a square, because each pair of adjacent This figure is a parallelogram, so it has rotational sides is congruent and perpendicular. symmetry of a half turn or 180 degrees around its All squares have both line and rotational symmetry. center, which is the point (1, -1.5). The line symmetry is vertically, horizontally, and diagonally through the center of the square, with lines Since this parallelogram is not a rhombus it does not that are either parallel to the sides of the square or have line symmetry. that include two vertices of the square. The equations of those lines are: x = 0, y = 0, y = x, and y ANSWER: = -x rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram The rotational symmetry is for each quarter turn in a onto itself. square, so the rotations of 90, 180, and 270 degrees around the origin map the square onto itself. 31. MODELING Symmetry is an important component of photography. Photographers often use reflection in water to create symmetry in photos. The photo on ANSWER: page 263 is a long exposure shot of the Eiffel tower line symmetry; rotational symmetry; the reflection in reflected in a pool. the line x = 0, the reflection in the line y = 0, the reflection in the line y = x, and the reflection in the a. line y = -x all map the square onto itself; the rotations Describe the two-dimensional symmetry created

of 90, 180, and 270 degrees around the origin map by the photo. the square onto itself. b. Is there rotational symmetry in the photo? Explain your reasoning. SOLUTION: a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. 29. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no SOLUTION: rotational symmetry. The trapezoid has line symmetry, because it is isosceles, but it does not have rotational symmetry, ANSWER: because no trapezoid does. a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. The reflection in the line y = 1.5 maps the trapezoid There is a vertical line of symmetry through the onto itself, because that is the perpendicular bisector center of the photo. to the parallel sides. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no ANSWER: rotational symmetry. line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself. COORDINATE GEOMETRY Determine whether the figure with the given vertices has line symmetry and/or rotational symmetry. 32. R(–3, 3), S(–3, –3), T(3, 3) SOLUTION: Draw the figure on a coordinate plane.

30. SOLUTION: This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its center, which is the point (1, -1.5).

31. MODELING Symmetry is an important component A figure in the plane has rotational symmetry if the of photography. Photographers often use reflection in figure can be mapped onto itself by a rotation water to create symmetry in photos. The photo on between 0° and 360° about the center of the figure. page 263 is a long exposure shot of the Eiffel tower There is not way to rotate the figure and have it map reflected in a pool. onto itself.

a. Describe the two-dimensional symmetry created Thus, the figure has only line symmetry. by the photo. ANSWER: b. Is there rotational symmetry in the photo? Explain line your reasoning.

SOLUTION: 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. SOLUTION: There is a vertical line of symmetry through the Draw the figure on a coordinate plane. center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no rotational symmetry.

ANSWER: a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. A figure has line symmetry if the figure can be b No; sample answer: Because of how the image is mapped onto itself by a reflection in a line. The reflected over the horizontal line, there is no given figure has 4 lines of symmetry. The line of rotational symmetry. symmetry are though the following pairs of points {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, COORDINATE GEOMETRY Determine and {(2, 2), (2, –2)}. whether the figure with the given vertices has line symmetry and/or rotational symmetry. A figure in the plane has rotational symmetry if the 32. R(–3, 3), S(–3, –3), T(3, 3) figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map Draw the figure on a coordinate plane. onto itself. The order of symmetry is 4.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 2) A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. SOLUTION: Draw the figure on a coordinate plane. The given triangle has a line of symmetry through points (0, 0) and (–3, 3).

onto itself. A figure has line symmetry if the figure can be Thus, the figure has only line symmetry. mapped onto itself by a reflection in a line. The given hexagon has 2 lines of symmetry. The lines ANSWER: pass through the following pair of points {(0, 4), (0, – line 4)}, and {(3, 0), (–3, 0)}

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational

35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) A figure has line symmetry if the figure can be SOLUTION: mapped onto itself by a reflection in a line. The Draw the figure on a coordinate plane. given figure has 4 lines of symmetry. The line of symmetry are though the following pairs of points {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, and {(2, 2), (2, –2)}.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4. A figure has line symmetry if the figure can be Thus, the figure has both line symmetry and mapped onto itself by a reflection in a line. The rotational symmetry. trapezoid has a line of reflection through points (0,3) and (0, –3). ANSWER: line and rotational A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – between 0° and 360° about the center of the 2) figure. There is no way to rotate this figure and have it map onto itself. Thus, it does not have rotational SOLUTION: symmetry. Draw the figure on a coordinate plane. Therefore, the figure has only line symmetry.

ANSWER: line

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: A figure has reflectional symmetry if the figure can line and rotational be mapped onto itself by a reflection in a line. The 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) line y = x has reflectional symmetry since any line perpendicular to y = x is a line of reflection. The SOLUTION: equation of the line symmetry is y = –x. Draw the figure on a coordinate plane. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The line can be rotated twice within 360° and be mapped onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational A figure has line symmetry if the figure can be symmetry. mapped onto itself by a reflection in a line. The trapezoid has a line of reflection through points (0,3) The magnitude of symmetry is the smallest angle and (0, –3). through which a figure can be rotated so that it maps onto itself. A figure in the plane has rotational symmetry if the The graph has magnitude of symmetry of figure can be mapped onto itself by a rotation . between 0° and 360° about the center of the figure. There is no way to rotate this figure and have Thus, the graph has both reflectional and rotational it map onto itself. Thus, it does not have rotational symmetry. symmetry. ANSWER: Therefore, the figure has only line symmetry. rotational; 2; 180°; line symmetry; y = –x ANSWER: line

SOLUTION: 2 37. y = x + 1 Graph the function. SOLUTION: Graph the function.

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The line y = x has reflectional symmetry since any line A figure has reflectional symmetry if the figure can perpendicular to y = x is a line of reflection. The be mapped onto itself by a reflection in a line. The equation of the line symmetry is y = –x. graph is reflected through the y-axis. Thus, the equation of the line symmetry is x = 0. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation A figure in the plane has rotational symmetry if the between 0° and 360° about the center of the figure. figure can be mapped onto itself by a rotation The line can be rotated twice within 360° and be between 0° and 360° about the center of the mapped onto itself. figure. There is no way to rotate the graph and have it map onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of Thus, the graph has only reflectional symmetry. symmetry. The graph has order 2 rotational symmetry. ANSWER: line; x = 0 The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of .

Thus, the graph has both reflectional and rotational symmetry.

ANSWER: 3 rotational; 2; 180°; line symmetry; y = –x 38. y = –x SOLUTION: Graph the function.

2 37. y = x + 1

Thus, the graph has only reflectional symmetry. ANSWER: rotational; 2; 180° ANSWER: line; x = 0

39. Refer to the rectangle on the coordinate plane.

38. y = –x3 SOLUTION: Graph the function.

Explain. A figure has reflectional symmetry if the figure can SOLUTION: be mapped onto itself by a reflection in a line. The a. The lines of symmetry are parallel to the sides of graph does not have a line of reflections where the graph can be mapped onto itself. the rectangles, and through the center of rotation.

A figure in the plane has rotational symmetry if the The slopes of the sides of the rectangle are 0.5 and figure can be mapped onto itself by a rotation -2, so the slopes of the lines of symmetry are the between 0° and 360° about the center of the same. figure. You can rotate the graph through the origin The center of the rectangle is (1, 1.5). Use the and have it map onto itself. point-slope formula to find equations.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto The magnitude of symmetry is the smallest angle itself under this rotation, the lines of symmetry are through which a figure can be rotated so that it maps mapped to each other. onto itself. The graph has magnitude of symmetry of . ANSWER: a. Thus, the graph has only rotational symmetry. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto ANSWER: itself under this rotation, the lines of symmetry are rotational; 2; 180° mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and

determine all possible lines of symmetry. Then record 39. Refer to the rectangle on the coordinate plane. their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. a. What are the equations of the lines of symmetry of d. Verbal Make a conjecture about the number of the rectangle? lines of symmetry and the order of symmetry for a b. What happens to the equations of the lines of regular polygon with n sides. symmetry when the rectangle is rotated 90 degrees SOLUTION: counterclockwise around its center of symmetry? a. Construct an equilateral triangle and label the Explain. vertices A, B, and C. Draw a line through A SOLUTION: perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image a. The lines of symmetry are parallel to the sides of maps to the original, then this line is a line of the rectangles, and through the center of rotation. reflection.

The slopes of the sides of the rectangle are 0.5 and -2, so the slopes of the lines of symmetry are the same. The center of the rectangle is (1, 1.5). Use the point-slope formula to find equations.

b. The equations of the lines of symmetry do not change; although the rectangle does not map onto Next, draw a line through B perpendicular to . itself under this rotation, the lines of symmetry are Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, mapped to each other. then this line is a line of reflection. ANSWER: a. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

then this line is a line of reflection. a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides.

SOLUTION: a. Construct an equilateral triangle and label the There are 3 lines of symmetry. vertices A, B, and C. Draw a line through A perpendicular to . Reflect the triangle in the line. b. Construct an equilateral triangle and show the Show the labels of the reflected image. If the image labels of the vertices. Next, find the center of the maps to the original, then this line is a line of triangle. Since this is an equilateral triangle, the reflection. circumcenter, incenter, centroid, and orthocenter are the same point. Construct altitudes through each vertex and label the intersection. Rotate the triangle about point D. A 120 degree rotation will map the image to the original. Show the labels of the image.

Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

There are 3 lines of symmetry.

b. Construct an equilateral triangle and show the labels of the vertices. Next, find the center of the triangle. Since this is an equilateral triangle, the Since the figure maps onto itself 3 times as it is circumcenter, incenter, centroid, and orthocenter are rotated, the order of symmetry is 3. the same point. Construct altitudes through each c. vertex and label the intersection. Square Rotate the triangle about point D. A 120 degree Construct a square and then construct lines through rotation will map the image to the original. Show the the midpoints of each side and diagonals. Use the labels of the image. reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5. The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

Regular Pentagon Construct a regular pentagon and then construct lines d. Sample answer: for each figure studied, the through each vertex perpendicular to the sides. Use number of sides of the figure is the same as the lines the reflection tool first to find that the image maps of symmetry and the order of symmetry. A regular onto the original when reflected in each of the 5 lines polygon with n sides has n lines of symmetry and constructed. So there are 5 lines of symmetry. order of symmetry n. Next, rotate the square about the center point. The ANSWER: image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5. a. 3 b. 3 c.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

Regular Hexagon Construct a regular hexagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps SOLUTION: onto the original when reflected in each of the 6 lines A figure has line symmetry if the figure can be constructed. So there are 6 lines of symmetry. mapped onto itself by a reflection in a line. This Next, rotate the square about the center point. The figure has 4 lines of symmetry. image maps to the original at 60, 120, 180, 240, 300, and 360 degree rotations. So the order of symmetry is 6.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure also has rotational symmetry.

ANSWER: 42. CHALLENGE A quadrilateral in the coordinate plane has exactly two lines of symmetry, y = x – 1 a. 3 and y = –x + 2. Find a set of possible vertices for b. 3 c. the figure. Graph the figure and the lines of symmetry. SOLUTION: Graph the figure and the lines of symmetry. d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

Pick points that are the same distance a from one line and the same distance b from the other line. In the same answer, the quadrilateral is a rectangle with sides which are parallel to the lines of symmetry. SOLUTION: This guarantees that the vertices of the quadrilateral A figure has line symmetry if the figure can be are the same distance a from one line and the same mapped onto itself by a reflection in a line. This distance b from the other line. In this case, a = figure has 4 lines of symmetry. and b = .

A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2).

ANSWER: Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2)

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure also has rotational symmetry. 43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. Therefore, neither of them are correct. Figure A has both line and rotational symmetry. SOLUTION: circle; Every line through the center of a circle is a ANSWER: line of symmetry, and there are infinitely many such Neither; Figure A has both line and rotational lines. symmetry. ANSWER: 42. CHALLENGE A quadrilateral in the coordinate circle; Every line through the center of a circle is a plane has exactly two lines of symmetry, y = x – 1 line of symmetry, and there are infinitely many such and y = –x + 2. Find a set of possible vertices for lines. the figure. Graph the figure and the lines of 44. OPEN-ENDED Draw a figure with line symmetry symmetry. but not rotational symmetry. Explain. SOLUTION: SOLUTION: Graph the figure and the lines of symmetry. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not have rotational symmetry.

and b = . ANSWER:

A set of possible vertices for the figure are, (–1, 0), Sample answer: An isosceles triangle has line (2, 3), (4, 1), and (1, 2). symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry ANSWER: because it cannot be rotated from 0° to 360° and Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) map onto itself.

45. WRITING IN MATH How are line symmetry and rotational symmetry related? SOLUTION: In both types of symmetries the figure is mapped 43. REASONING A figure has infinitely many lines of onto itself. symmetry. What is the figure? Explain. SOLUTION: Rotational symmetry. circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines. Reflectional symmetry:

ANSWER: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. In some cases an object can have both rotational and reflectional symmetry, such as the diamond, however SOLUTION: some objects do not have both such as the crab. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not have rotational symmetry.

ANSWER: An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does Sample answer: In both rotational and line symmetry not have rotational symmetry because it cannot be a figure is mapped onto itself. However, in line rotated from 0° to 360° and map onto itself. symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry.

A 2 B 3 C 45. WRITING IN MATH How are line symmetry and 4 rotational symmetry related? D 8 SOLUTION: SOLUTION: In both types of symmetries the figure is mapped onto itself.

Rotational symmetry.

2 + 2 = 4, so C is the correct answer.

ANSWER: In some cases an object can have both rotational and reflectional symmetry, such as the diamond, however C some objects do not have both such as the crab. 47. Patrick drew a figure that has rotational symmetry

but not line symmetry. Which of the following could be the figure that Patrick drew? A

ANSWER: Sample answer: In both rotational and line symmetry C a figure is mapped onto itself. However, in line symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have D line symmetry and rotational symmetry.

46. Sasha owns a tile store. For each tile in her store, she

calculates the sum of the number of lines of E symmetry and the order of symmetry, and then she enters this value into a database. Which value should she enter in the database for the tile shown here? SOLUTION:

Option A has rotational and reflectional symmetry.

Option B has reflectional symmetry but not rotational A 2 symmetry. B 3 C 4 D 8 SOLUTION:

Option C has neither rotational nor reflectional symmetry.

2 + 2 = 4, so C is the correct answer.

ANSWER: Option E has reflectional symmetry but not rotational symmetry. C

47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A

The correct choice is D. B ANSWER: D

C 48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle D B Equiangular triangle C Isosceles triangle D Scalene triangle E SOLUTION: An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

SOLUTION: Option A has rotational and reflectional symmetry. ANSWER: C

Option D has rotational symmetry but not reflectional symmetry.

Option E has reflectional symmetry but not rotational symmetry. Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

The correct choice is D.

ANSWER: D

48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle D Scalene triangle Option B has reflective symmetry but not rotational SOLUTION: symmetry. The correct choice is B. An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D SOLUTION: First, plot the points.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

Thus, there are four possible lines that go through the center and are lines of reflections.

State whether the figure appears to have line Therefore, the figure has four lines of symmetry. symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state ANSWER: their number. yes; 4

1. SOLUTION:

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of 2. reflection must go through the center point. SOLUTION: Two lines of reflection go through the sides of the A figure has reflectional symmetry if the figure can figure. be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER: no

Two lines of reflection go through the vertices of the figure.

3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry. Thus, there are four possible lines that go through the center and are lines of reflections. The figure has a vertical line of symmetry.

It does not have a horizontal line of symmetry. Therefore, the figure has four lines of symmetry.

ANSWER: yes; 4

The figure does not have a line of symmetry through the vertices.

Thus, the figure has only one line of symmetry. 2. ANSWER: SOLUTION: yes; 1 A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER: State whether the figure has rotational no symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

SOLUTION: 4. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. SOLUTION: A figure in the plane has rotational symmetry if the The given figure has reflectional symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure has a vertical line of symmetry. For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: no

It does not have a horizontal line of symmetry.

The figure does not have a line of symmetry through SOLUTION: the vertices. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

Thus, the figure has only one line of symmetry.

ANSWER: yes; 1

The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of symmetry.

The given figure has order of symmetry of 2, since State whether the figure has rotational the figure can be rotated twice in 360°. symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state The magnitude of symmetry is the smallest angle the order and magnitude of symmetry. through which a figure can be rotated so that it maps onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is . 4. SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the yes; 2; 180° figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure

were a regular pentagon, it would have rotational symmetry.

ANSWER: no 6. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

5. SOLUTION: The given figure has rotational symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

The number of times a figure maps onto itself as it The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps rotates form 0° and 360° is called the order of onto itself. symmetry.

The figure has magnitude of symmetry of The given figure has order of symmetry of 2, since . the figure can be rotated twice in 360°. ANSWER: The magnitude of symmetry is the smallest angle yes; 4; 90° through which a figure can be rotated so that it maps onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is . ANSWER: yes; 2; 180° State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

6. 7. SOLUTION: SOLUTION: A figure in the plane has rotational symmetry if the Vertical and horizontal lines through the center and figure can be mapped onto itself by a rotation diagonal lines through two opposite vertices are all between 0° and 360° about the center of the figure. lines of symmetry for a square oriented this way.

The given figure has rotational symmetry. The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1.

Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

ANSWER: line symmetry; rotational symmetry; the reflection in The number of times a figure maps onto itself as it the line x = 0, the reflection in the line y = -1, the rotates from 0° to 360° is called the order of reflection in the line y = x - 1, and the reflection in symmetry. the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point Since the figure can be rotated 4 times within 360° , (0, -1) map the square onto itself. it has order 4 rotational symmetry

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of . 8. ANSWER: SOLUTION: yes; 4; 90° This figure does not have line symmetry, because adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

ANSWER: State whether the figure has line symmetry rotational symmetry; the rotation of 180 degrees and/or rotational symmetry. If so, describe the around the point (1, 1) maps the parallelogram onto reflections and/or rotations that map the figure itself. onto itself. REGULARITY State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

7. 9. SOLUTION: SOLUTION: Vertical and horizontal lines through the center and diagonal lines through two opposite vertices are all A figure has reflectional symmetry if the figure can lines of symmetry for a square oriented this way. be mapped onto itself by a reflection in a line.

The equations of those lines in this figure are x = 0, For the given figure, there are no lines of reflection y = -1, y = x - 1, and y = -x - 1. where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry. Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around ANSWER: the point (0, -1) map the square onto itself. no

ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the 10. rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

8. The figure has a vertical and horizontal line of SOLUTION: reflection. This figure does not have line symmetry, because adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

ANSWER: It is also possible to have reflection over the rotational symmetry; the rotation of 180 degrees diagonal lines. around the point (1, 1) maps the parallelogram onto itself.

REGULARITY State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number. Therefore, the figure has four lines of symmetry

9. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection ANSWER: where the figure can map onto itself. Thus, the figure yes; 4 does not have any lines of of symmetry.

ANSWER: no

10. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line. 11.

The given figure has reflectional symmetry. SOLUTION: A figure has reflectional symmetry if the figure can In order for the figure to map onto itself, the line of be mapped onto itself by a reflection in a line. reflection must go through the center point. The given hexagon has reflectional symmetry. The figure has a vertical and horizontal line of reflection. In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though opposites edges.

It is also possible to have reflection over the diagonal lines.

There are three lines of reflection that go though opposites vertices.

Therefore, the figure has four lines of symmetry

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry.

ANSWER: yes; 4

ANSWER: yes; 6

11.

SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

12. The given hexagon has reflectional symmetry. SOLUTION: In order for the hexagon to map onto itself, the line A figure has reflectional symmetry if the figure can of reflection must go through the center point. be mapped onto itself by a reflection in a line.

There are three lines of reflection that go though The figure has reflectional symmetry. opposites edges. There is only one line of symmetry, a horizontal line through the middle of the figure.

There are three lines of reflection that go though Thus, the figure has one line of symmetry. opposites vertices. ANSWER: yes; 1

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry. 13. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

There is only one possible line of reflection, horizontally though the middle of the figure. ANSWER: yes; 6

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

3-5 Symmetry

12. 14. SOLUTION: A figure has reflectional symmetry if the figure can SOLUTION: be mapped onto itself by a reflection in a line. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The figure has reflectional symmetry. The given figure does not have reflectional There is only one line of symmetry, a horizontal line symmetry. It is not possible to draw a line of through the middle of the figure. reflection where the figure can map onto itself. ANSWER: no

FLAGS State whether each flag design appears to Thus, the figure has one line of symmetry. have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their ANSWER: number. yes; 1 15. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

SOLUTION: ANSWER: A figure has reflectional symmetry if the figure can no be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. 16. Refer to the flag on page 262. SOLUTION: There is only one possible line of reflection, A figure has reflectional symmetry if the figure can horizontally though the middle of the figure. be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

Thus, the figure has one line of symmetry. In order for the figure to map onto itself, the line of reflection must go through the center point. ANSWER: yes; 1 A horizontal and vertical lines of reflection are possible.

eSolutions14. Manual - Powered by Cognero Page 6 SOLUTION: A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line. Two diagonal lines of reflection are possible.

The given figure does not have reflectional symmetry. It is not possible to draw a line of reflection where the figure can map onto itself.

ANSWER: no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their number. There are a total of four possible lines that go 15. Refer to page 262. through the center and are lines of reflections. Thus, the flag has four lines of symmetry. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER: no ANSWER: 16. Refer to the flag on page 262. yes; 4 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible. 17. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

Two diagonal lines of reflection are possible.

It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry.

ANSWER:

yes; 1 There are a total of four possible lines that go through the center and are lines of reflections. Thus, the flag has four lines of symmetry.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

18. ANSWER: yes; 4 SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

17. Refer to page 262. The number of times a figure maps onto itself as it SOLUTION: rotates from 0° to 360° is called the order of

A figure has reflectional symmetry if the figure can symmetry. be mapped onto itself by a reflection in a line. This figure has order 2 rotational symmetry, since The figure has reflectional symmetry. you have to rotate 180° to get the figure to map onto

itself.

A horizontal line is a line of reflections for this flag. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has a magnitude of symmetry of . It is not possible to reflect over a vertical or line ANSWER: through the diagonals.

Thus, the figure has one line of symmetry.

ANSWER:

yes; 1 yes; 2; 180°

19. State whether the figure has rotational SOLUTION: symmetry. Write yes or no. If so, copy the A figure in the plane has rotational symmetry if the figure, locate the center of symmetry, and state figure can be mapped onto itself by a rotation the order and magnitude of symmetry. between 0° and 360° about the center of the figure.

The triangle has rotational symmetry.

18. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 3 rotational symmetry. The figure has rotational symmetry.

The figure has magnitude of symmetry of This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto .

itself. ANSWER:

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has a magnitude of symmetry of yes; 3; 120° .

ANSWER:

20. SOLUTION: A figure in the plane has rotational symmetry if the

yes; 2; 180° figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be 19. mapped onto itself.

SOLUTION: A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation no between 0° and 360° about the center of the figure.

The triangle has rotational symmetry.

21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 3; 120°

20.

SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation ANSWER: between 0° and 360° about the center of the figure. no

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

22. ANSWER: SOLUTION: no A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry. 21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 8; 45°

ANSWER: no 23. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. 22. SOLUTION: The figure has rotational symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This means that the figure can be rotated 8 times and map

onto itself within 360°. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The magnitude of symmetry is the smallest angle symmetry. through which a figure can be rotated so that it maps onto itself. The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it The figure has magnitude of symmetry of map onto itself within 360°. .

ANSWER: The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of . yes; 8; 45°

ANSWER: WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, state the order and magnitude of symmetry. 24. Refer to page 263. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation yes; 8; 45° between 0° and 360° about the center of the figure.

The wheel has rotational symmetry.

The number of times a figure maps onto itself as it 23. rotates from 0° to 360° is called the order of SOLUTION: symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The wheel has order 5 rotational symmetry. There between 0° and 360° about the center of the figure. are 5 large spokes and 5 small spokes. You can rotate the wheel 5 times within 360° and map the figure onto itself. The figure has rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The wheel has magnitude of symmetry . The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of ANSWER: symmetry. yes; 5; 72°

The figure has order 8 rotational symmetry. This 25. Refer to page 263. means that the figure can be rotated 8 times and map onto itself within 360°. SOLUTION: A figure in the plane has rotational symmetry if the The magnitude of symmetry is the smallest angle figure can be mapped onto itself by a rotation through which a figure can be rotated so that it maps between 0° and 360° about the center of the figure. onto itself. The wheel has rotational symmetry. The figure has magnitude of symmetry of . The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of ANSWER: symmetry.

The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself.

yes; 8; 45° The magnitude of symmetry is the smallest angle WHEELS State whether each wheel cover appears through which a figure can be rotated so that it maps to have rotational symmetry. Write yes or no. If so, onto itself. state the order and magnitude of symmetry. 24. Refer to page 263. The wheel has order 8 rotational symmetry and magnitude . SOLUTION: A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation yes; 8; 45° between 0° and 360° about the center of the figure. 26. Refer to page 263.

The wheel has rotational symmetry. SOLUTION:

The wheel has order 5 rotational symmetry. There The number of times a figure maps onto itself as it are 5 large spokes and 5 small spokes. You can rotates from 0° to 360° is called the order of rotate the wheel 5 times within 360° and map the symmetry. The wheel has order 10 rotational figure onto itself. symmetry. There are 10 bolts and the tire can be rotated 10 times within 360° and map onto itself. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The magnitude of symmetry is the smallest angle onto itself. through which a figure can be rotated so that it maps onto itself. The wheel has magnitude of symmetry of The wheel has magnitude of symmetry . . ANSWER: ANSWER: yes; 10; 36° yes; 5; 72° State whether the figure has line symmetry 25. Refer to page 263. and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure SOLUTION: onto itself. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The wheel has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of 27. symmetry. SOLUTION:

The wheel has order 8 rotational symmetry. There This triangle is scalene, so it cannot have symmetry. are 8 spokes, thus the wheel can be rotated 8 times ANSWER: within 360° and map onto itself. no symmetry The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The wheel has order 8 rotational symmetry and magnitude .

ANSWER: 28. yes; 8; 45° SOLUTION: 26. Refer to page 263. This figure is a square, because each pair of adjacent sides is congruent and perpendicular. SOLUTION: All squares have both line and rotational symmetry. A figure in the plane has rotational symmetry if the The line symmetry is vertically, horizontally, and figure can be mapped onto itself by a rotation diagonally through the center of the square, with lines between 0° and 360° about the center of the figure. that are either parallel to the sides of the square or The wheel has rotational symmetry. that include two vertices of the square. The equations of those lines are: x = 0, y = 0, y = x, and y The number of times a figure maps onto itself as it = -x rotates from 0° to 360° is called the order of symmetry. The wheel has order 10 rotational The rotational symmetry is for each quarter turn in a symmetry. There are 10 bolts and the tire can be square, so the rotations of 90, 180, and 270 degrees rotated 10 times within 360° and map onto itself. around the origin map the square onto itself.

The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps line symmetry; rotational symmetry; the reflection in onto itself. The wheel has magnitude of symmetry of the line x = 0, the reflection in the line y = 0, the . reflection in the line y = x, and the reflection in the ANSWER: line y = -x all map the square onto itself; the rotations of 90, 180, and 270 degrees around the origin map yes; 10; 36° the square onto itself. State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

29. SOLUTION: The trapezoid has line symmetry, because it is isosceles, but it does not have rotational symmetry, 27. because no trapezoid does. SOLUTION: The reflection in the line y = 1.5 maps the trapezoid This triangle is scalene, so it cannot have symmetry. onto itself, because that is the perpendicular bisector ANSWER: to the parallel sides. no symmetry ANSWER: line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself.

28. SOLUTION: This figure is a square, because each pair of adjacent 30. sides is congruent and perpendicular. SOLUTION: All squares have both line and rotational symmetry. This figure is a parallelogram, so it has rotational The line symmetry is vertically, horizontally, and symmetry of a half turn or 180 degrees around its diagonally through the center of the square, with lines center, which is the point (1, -1.5). that are either parallel to the sides of the square or

that include two vertices of the square. The Since this parallelogram is not a rhombus it does not equations of those lines are: x = 0, y = 0, y = x, and y have line symmetry. = -x ANSWER: The rotational symmetry is for each quarter turn in a rotational symmetry; the rotation of 180 degrees square, so the rotations of 90, 180, and 270 degrees around the point (1, -1.5) maps the parallelogram around the origin map the square onto itself. onto itself.

ANSWER: 31. MODELING Symmetry is an important component of photography. Photographers often use reflection in line symmetry; rotational symmetry; the reflection in water to create symmetry in photos. The photo on the line x = 0, the reflection in the line y = 0, the page 263 is a long exposure shot of the Eiffel tower reflection in the line y = x, and the reflection in the reflected in a pool. line y = -x all map the square onto itself; the rotations

of 90, 180, and 270 degrees around the origin map the square onto itself. a. Describe the two-dimensional symmetry created by the photo. b. Is there rotational symmetry in the photo? Explain your reasoning. SOLUTION: a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. 29. There is a vertical line of symmetry through the center of the photo. SOLUTION: b No; sample answer: Because of how the image is The trapezoid has line symmetry, because it is reflected over the horizontal line, there is no isosceles, but it does not have rotational symmetry, rotational symmetry. because no trapezoid does. ANSWER: The reflection in the line y = 1.5 maps the trapezoid a. Sample answer: There is a horizontal line of onto itself, because that is the perpendicular bisector symmetry between the tower and its reflection. to the parallel sides. There is a vertical line of symmetry through the center of the photo. ANSWER: b No; sample answer: Because of how the image is line symmetry; the reflection in the line y = 1.5 maps reflected over the horizontal line, there is no the trapezoid onto itself. rotational symmetry.

COORDINATE GEOMETRY Determine whether the figure with the given vertices has line symmetry and/or rotational symmetry. 32. R(–3, 3), S(–3, –3), T(3, 3) SOLUTION: Draw the figure on a coordinate plane. 30. SOLUTION: This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its center, which is the point (1, -1.5).

Since this parallelogram is not a rhombus it does not have line symmetry.

onto itself. The given triangle has a line of symmetry through 31. MODELING Symmetry is an important component points (0, 0) and (–3, 3). of photography. Photographers often use reflection in water to create symmetry in photos. The photo on A figure in the plane has rotational symmetry if the page 263 is a long exposure shot of the Eiffel tower figure can be mapped onto itself by a rotation reflected in a pool. between 0° and 360° about the center of the figure. There is not way to rotate the figure and have it map

a. Describe the two-dimensional symmetry created onto itself. by the photo. b. Is there rotational symmetry in the photo? Explain Thus, the figure has only line symmetry. your reasoning. ANSWER: SOLUTION: line a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) There is a vertical line of symmetry through the center of the photo. SOLUTION: b No; sample answer: Because of how the image is Draw the figure on a coordinate plane. reflected over the horizontal line, there is no rotational symmetry.

ANSWER: a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no A figure has line symmetry if the figure can be rotational symmetry. mapped onto itself by a reflection in a line. The given figure has 4 lines of symmetry. The line of COORDINATE GEOMETRY Determine symmetry are though the following pairs of points whether the figure with the given vertices has line {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, symmetry and/or rotational symmetry. and {(2, 2), (2, –2)}. 32. R(–3, 3), S(–3, –3), T(3, 3) SOLUTION: A figure in the plane has rotational symmetry if the Draw the figure on a coordinate plane. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational

A figure has line symmetry if the figure can be 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – mapped onto itself by a reflection in a line. 2)

The given triangle has a line of symmetry through SOLUTION: points (0, 0) and (–3, 3). Draw the figure on a coordinate plane.

SOLUTION: A figure in the plane has rotational symmetry if the Draw the figure on a coordinate plane. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure has rotational symmetry. You can rotate the figure once within 360° and have it map to itself.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) given figure has 4 lines of symmetry. The line of SOLUTION: symmetry are though the following pairs of points Draw the figure on a coordinate plane. {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, and {(2, 2), (2, –2)}.

Thus, the figure has both line symmetry and rotational symmetry. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The ANSWER: trapezoid has a line of reflection through points (0,3) line and rotational and (0, –3).

34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – A figure in the plane has rotational symmetry if the 2) figure can be mapped onto itself by a rotation between 0° and 360° about the center of the SOLUTION: figure. There is no way to rotate this figure and have Draw the figure on a coordinate plane. it map onto itself. Thus, it does not have rotational symmetry.

Therefore, the figure has only line symmetry.

ANSWER: line

ALGEBRA Graph the function and determine whether the graph has line and/or rotational A figure has line symmetry if the figure can be symmetry. If so, state the order and magnitude of mapped onto itself by a reflection in a line. The symmetry, and write the equations of any lines of given hexagon has 2 lines of symmetry. The lines symmetry. pass through the following pair of points {(0, 4), (0, – 36. y = x 4)}, and {(3, 0), (–3, 0)} SOLUTION: A figure in the plane has rotational symmetry if the Graph the function. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure has rotational symmetry. You can rotate the figure once within 360° and have it map to itself.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER:

line and rotational 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The SOLUTION: line y = x has reflectional symmetry since any line Draw the figure on a coordinate plane. perpendicular to y = x is a line of reflection. The equation of the line symmetry is y = –x.

The number of times a figure maps onto itself as it A figure has line symmetry if the figure can be rotates from 0° to 360° is called the order of mapped onto itself by a reflection in a line. The symmetry. The graph has order 2 rotational trapezoid has a line of reflection through points (0,3) symmetry. and (0, –3). The magnitude of symmetry is the smallest angle A figure in the plane has rotational symmetry if the through which a figure can be rotated so that it maps figure can be mapped onto itself by a rotation onto itself. between 0° and 360° about the center of the The graph has magnitude of symmetry of figure. There is no way to rotate this figure and have . it map onto itself. Thus, it does not have rotational symmetry. Thus, the graph has both reflectional and rotational symmetry. Therefore, the figure has only line symmetry. ANSWER: ANSWER: rotational; 2; 180°; line symmetry; y = –x line

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The line y = x has reflectional symmetry since any line perpendicular to y = x is a line of reflection. The equation of the line symmetry is y = –x. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The A figure in the plane has rotational symmetry if the graph is reflected through the y-axis. Thus, the figure can be mapped onto itself by a rotation equation of the line symmetry is x = 0. between 0° and 360° about the center of the figure. The line can be rotated twice within 360° and be A figure in the plane has rotational symmetry if the mapped onto itself. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the The number of times a figure maps onto itself as it figure. There is no way to rotate the graph and have rotates from 0° to 360° is called the order of it map onto itself. symmetry. The graph has order 2 rotational symmetry. Thus, the graph has only reflectional symmetry.

The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps line; x = 0 onto itself. The graph has magnitude of symmetry of .

Thus, the graph has both reflectional and rotational symmetry.

ANSWER: rotational; 2; 180°; line symmetry; y = –x 38. y = –x3 SOLUTION: Graph the function.

2 37. y = x + 1

SOLUTION: Graph the function. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. You can rotate the graph through the origin and have it map onto itself. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The The number of times a figure maps onto itself as it graph is reflected through the y-axis. Thus, the rotates from 0° to 360° is called the order of equation of the line symmetry is x = 0. symmetry. The graph has order 2 rotational symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The magnitude of symmetry is the smallest angle between 0° and 360° about the center of the through which a figure can be rotated so that it maps figure. There is no way to rotate the graph and have onto itself. The graph has magnitude of symmetry of it map onto itself. .

Thus, the graph has only reflectional symmetry. Thus, the graph has only rotational symmetry.

ANSWER: ANSWER: line; x = 0 rotational; 2; 180°

3 38. y = –x 39. Refer to the rectangle on the coordinate plane. SOLUTION: Graph the function.

a. What are the equations of the lines of symmetry of the rectangle? b. What happens to the equations of the lines of

symmetry when the rectangle is rotated 90 degrees A figure has reflectional symmetry if the figure can counterclockwise around its center of symmetry? be mapped onto itself by a reflection in a line. The Explain. graph does not have a line of reflections where the SOLUTION: graph can be mapped onto itself. a. The lines of symmetry are parallel to the sides of

A figure in the plane has rotational symmetry if the the rectangles, and through the center of rotation. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the The slopes of the sides of the rectangle are 0.5 and figure. You can rotate the graph through the origin -2, so the slopes of the lines of symmetry are the and have it map onto itself. same. The center of the rectangle is (1, 1.5). Use the The number of times a figure maps onto itself as it point-slope formula to find equations. rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry.

ANSWER: Thus, the graph has only rotational symmetry. a. ANSWER: b. The equations of the lines of symmetry do not rotational; 2; 180° change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool 39. Refer to the rectangle on the coordinate plane. under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. a. What are the equations of the lines of symmetry of Record the number of lines of symmetry and the the rectangle? order of symmetry for each polygon. b. What happens to the equations of the lines of d. Verbal Make a conjecture about the number of symmetry when the rectangle is rotated 90 degrees lines of symmetry and the order of symmetry for a regular polygon with n sides. counterclockwise around its center of symmetry? Explain. SOLUTION: a. Construct an equilateral triangle and label the SOLUTION: vertices A, B, and C. Draw a line through A a. The lines of symmetry are parallel to the sides of perpendicular to . Reflect the triangle in the line. the rectangles, and through the center of rotation. Show the labels of the reflected image. If the image maps to the original, then this line is a line of The slopes of the sides of the rectangle are 0.5 and reflection. -2, so the slopes of the lines of symmetry are the same. The center of the rectangle is (1, 1.5). Use the point-slope formula to find equations.

b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons. Lastly, draw a line through C perpendicular to . a. Geometric Use The Geometer’s Sketchpad to Reflect the triangle in the line. Show the labels of the draw an equilateral triangle. Use the reflection tool reflected image. If the image maps to the original, under the transformation menu to investigate and then this line is a line of reflection. determine all possible lines of symmetry. Then record their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides. SOLUTION: a. Construct an equilateral triangle and label the vertices A, B, and C. Draw a line through A perpendicular to . Reflect the triangle in the line. There are 3 lines of symmetry. Show the labels of the reflected image. If the image maps to the original, then this line is a line of b. Construct an equilateral triangle and show the reflection. labels of the vertices. Next, find the center of the triangle. Since this is an equilateral triangle, the circumcenter, incenter, centroid, and orthocenter are the same point. Construct altitudes through each vertex and label the intersection. Rotate the triangle about point D. A 120 degree rotation will map the image to the original. Show the labels of the image.

Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

There are 3 lines of symmetry.

b. Construct an equilateral triangle and show the labels of the vertices. Next, find the center of the triangle. Since this is an equilateral triangle, the circumcenter, incenter, centroid, and orthocenter are the same point. Construct altitudes through each vertex and label the intersection. Since the figure maps onto itself 3 times as it is Rotate the triangle about point D. A 120 degree rotated, the order of symmetry is 3. rotation will map the image to the original. Show the c. labels of the image. Square Construct a square and then construct lines through the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The The triangle can be rotated a third time about D. A image maps to the original at 72, 144, 216, 288, and 360 degree rotation maps the image to the original. 360 degree rotations. So the order of symmetry is 5.

Since the figure maps onto itself 3 times as it is rotated, the order of symmetry is 3. c. Regular Hexagon Square Construct a regular hexagon and then construct lines Construct a square and then construct lines through through each vertex perpendicular to the sides. Use the midpoints of each side and diagonals. Use the the reflection tool first to find that the image maps reflection tool first to find that the image maps onto onto the original when reflected in each of the 6 lines the original when reflected in each of the 4 lines constructed. So there are 6 lines of symmetry. constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The Next, rotate the square about the center point. The image maps to the original at 60, 120, 180, 240, 300, image maps to the original at 90, 180, 270, and 360 and 360 degree rotations. So the order of symmetry degree rotations. So the order of symmetry is 4. is 6.

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps d. Sample answer: for each figure studied, the onto the original when reflected in each of the 5 lines number of sides of the figure is the same as the lines constructed. So there are 5 lines of symmetry. of symmetry and the order of symmetry. A regular Next, rotate the square about the center point. The polygon with n sides has n lines of symmetry and image maps to the original at 72, 144, 216, 288, and order of symmetry n. 360 degree rotations. So the order of symmetry is 5. ANSWER:

a. 3 b. 3 c.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure also has rotational symmetry.

d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines Therefore, neither of them are correct. Figure A has of symmetry and the order of symmetry. A regular both line and rotational symmetry. polygon with n sides has n lines of symmetry and ANSWER: order of symmetry n. Neither; Figure A has both line and rotational ANSWER: symmetry. a. 3 42. CHALLENGE A quadrilateral in the coordinate b. 3 plane has exactly two lines of symmetry, y = x – 1 c. and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of symmetry.

d. Sample answer: A regular polygon with n sides SOLUTION: has n lines of symmetry and order of symmetry n. Graph the figure and the lines of symmetry.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

Pick points that are the same distance a from one line and the same distance b from the other line. In SOLUTION: the same answer, the quadrilateral is a rectangle with A figure has line symmetry if the figure can be sides which are parallel to the lines of symmetry. mapped onto itself by a reflection in a line. This This guarantees that the vertices of the quadrilateral figure has 4 lines of symmetry. are the same distance a from one line and the same distance b from the other line. In this case, a =

and b = .

A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2).

ANSWER: A figure in the plane has rotational symmetry if the Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure also has rotational symmetry.

Therefore, neither of them are correct. Figure A has 43. REASONING A figure has infinitely many lines of both line and rotational symmetry. symmetry. What is the figure? Explain. ANSWER: SOLUTION: Neither; Figure A has both line and rotational circle; Every line through the center of a circle is a symmetry. line of symmetry, and there are infinitely many such lines. 42. CHALLENGE A quadrilateral in the coordinate plane has exactly two lines of symmetry, y = x – 1 ANSWER: and y = –x + 2. Find a set of possible vertices for circle; Every line through the center of a circle is a the figure. Graph the figure and the lines of line of symmetry, and there are infinitely many such

symmetry. lines. SOLUTION: 44. OPEN-ENDED Draw a figure with line symmetry Graph the figure and the lines of symmetry. but not rotational symmetry. Explain. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not have rotational symmetry. Pick points that are the same distance a from one line and the same distance b from the other line. In An isosceles triangle has line symmetry from the the same answer, the quadrilateral is a rectangle with vertex angle to the base of the triangle, but it does sides which are parallel to the lines of symmetry. not have rotational symmetry because it cannot be This guarantees that the vertices of the quadrilateral rotated from 0° to 360° and map onto itself. are the same distance a from one line and the same distance b from the other line. In this case, a =

and b = .

A set of possible vertices for the figure are, (–1, 0), ANSWER: (2, 3), (4, 1), and (1, 2). Sample answer: An isosceles triangle has line ANSWER: symmetry from the vertex angle to the base of the Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

45. WRITING IN MATH How are line symmetry and rotational symmetry related? 43. REASONING A figure has infinitely many lines of SOLUTION: symmetry. What is the figure? Explain. In both types of symmetries the figure is mapped SOLUTION: onto itself. circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such Rotational symmetry. lines.

ANSWER: Reflectional symmetry: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain.

SOLUTION: In some cases an object can have both rotational and A figure has line symmetry if the figure can be reflectional symmetry, such as the diamond, however mapped onto itself by a reflection in a line. A figure some objects do not have both such as the crab. in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not have rotational symmetry.

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does ANSWER: not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself. Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry.

46. Sasha owns a tile store. For each tile in her store, she ANSWER: calculates the sum of the number of lines of Sample answer: An isosceles triangle has line symmetry and the order of symmetry, and then she symmetry from the vertex angle to the base of the enters this value into a database. Which value should triangle, but it does not have rotational symmetry she enter in the database for the tile shown here? because it cannot be rotated from 0° to 360° and map onto itself.

A 45. WRITING IN MATH How are line symmetry and 2 rotational symmetry related? B 3 C 4 SOLUTION: D 8 In both types of symmetries the figure is mapped onto itself. SOLUTION:

Rotational symmetry.

Reflectional symmetry: The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn around its center.

In some cases an object can have both rotational and 2 + 2 = 4, so C is the correct answer. reflectional symmetry, such as the diamond, however some objects do not have both such as the crab. ANSWER: C 47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A

ANSWER: B Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is C mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry. D 46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of symmetry and the order of symmetry, and then she enters this value into a database. Which value should E she enter in the database for the tile shown here?

SOLUTION: Option A has rotational and reflectional symmetry.

A 2 B 3 Option B has reflectional symmetry but not rotational C 4 symmetry. D 8 SOLUTION:

Option C has neither rotational nor reflectional symmetry.

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has Option D has rotational symmetry but not reflectional rotational symmetry at 180 degrees, or each half turn symmetry. around its center.

2 + 2 = 4, so C is the correct answer.

ANSWER: C Option E has reflectional symmetry but not rotational 47. Patrick drew a figure that has rotational symmetry symmetry. but not line symmetry. Which of the following could be the figure that Patrick drew? A

The correct choice is D.

ANSWER: C D

48. Which of the following figures may have exactly one D line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle E D Scalene triangle SOLUTION: An isosceles triangle has one line of symmetry and SOLUTION: no rotational symmetry. The correct choice is C. Option A has rotational and reflectional symmetry.

ANSWER: C

49. Camryn plotted the points , Option B has reflectional symmetry but not rotational and . Which of the following additional points symmetry. can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C

Option C has neither rotational nor reflectional D symmetry. SOLUTION: First, plot the points.

Option D has rotational symmetry but not reflectional symmetry.

Option E has reflectional symmetry but not rotational symmetry.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

The correct choice is D.

ANSWER: D

48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle D Scalene triangle

SOLUTION: An isosceles triangle has one line of symmetry and Option B has reflective symmetry but not rotational no rotational symmetry. The correct choice is C. symmetry. The correct choice is B.

ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D SOLUTION: First, plot the points.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

Thus, there are four possible lines that go through the center and are lines of reflections.

State whether the figure appears to have line symmetry. Write yes or no. If so, copy the Therefore, the figure has four lines of symmetry. figure, draw all lines of symmetry, and state their number. ANSWER: yes; 4

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point. 2. SOLUTION: Two lines of reflection go through the sides of the A figure has reflectional symmetry if the figure can figure. be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER:

no Two lines of reflection go through the vertices of the figure.

3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

Thus, there are four possible lines that go through The given figure has reflectional symmetry. the center and are lines of reflections. The figure has a vertical line of symmetry.

It does not have a horizontal line of symmetry. Therefore, the figure has four lines of symmetry.

ANSWER: yes; 4

The figure does not have a line of symmetry through the vertices.

2. Thus, the figure has only one line of symmetry. SOLUTION: ANSWER: A figure has reflectional symmetry if the figure can yes; 1 be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER: no State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

3. SOLUTION: A figure has reflectional symmetry if the figure can 4. be mapped onto itself by a reflection in a line. SOLUTION:

A figure in the plane has rotational symmetry if the The given figure has reflectional symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure has a vertical line of symmetry. For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER:

no It does not have a horizontal line of symmetry.

5. The figure does not have a line of symmetry through SOLUTION: the vertices. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

Thus, the figure has only one line of symmetry.

ANSWER: yes; 1

The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of symmetry.

State whether the figure has rotational The given figure has order of symmetry of 2, since symmetry. Write yes or no. If so, copy the the figure can be rotated twice in 360°. figure, locate the center of symmetry, and state the order and magnitude of symmetry. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

Since the figure has order 2 rotational symmetry, the 4. magnitude of the symmetry is .

SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the yes; 2; 180° figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: no 6. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 5. between 0° and 360° about the center of the figure.

SOLUTION: The given figure has rotational symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

The number of times a figure maps onto itself as it The magnitude of symmetry is the smallest angle rotates form 0° and 360° is called the order of through which a figure can be rotated so that it maps symmetry. onto itself.

The given figure has order of symmetry of 2, since The figure has magnitude of symmetry of the figure can be rotated twice in 360°. .

The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps yes; 4; 90° onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is .

ANSWER:

yes; 2; 180° State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

between 0° and 360° about the center of the figure. lines of symmetry for a square oriented this way.

The given figure has rotational symmetry. The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1.

Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

ANSWER:

The number of times a figure maps onto itself as it line symmetry; rotational symmetry; the reflection in rotates from 0° to 360° is called the order of the line x = 0, the reflection in the line y = -1, the symmetry. reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point Since the figure can be rotated 4 times within 360° , (0, -1) map the square onto itself. it has order 4 rotational symmetry

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of . 8. ANSWER: yes; 4; 90° SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

7. SOLUTION: 9. Vertical and horizontal lines through the center and SOLUTION: diagonal lines through two opposite vertices are all A figure has reflectional symmetry if the figure can lines of symmetry for a square oriented this way. be mapped onto itself by a reflection in a line.

The equations of those lines in this figure are x = 0, For the given figure, there are no lines of reflection y = -1, y = x - 1, and y = -x - 1. where the figure can map onto itself. Thus, the figure

does not have any lines of of symmetry. Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around ANSWER: the point (0, -1) map the square onto itself. no ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point 10. (0, -1) map the square onto itself. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point. 8. The figure has a vertical and horizontal line of SOLUTION: reflection. This figure does not have line symmetry, because adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

ANSWER: rotational symmetry; the rotation of 180 degrees It is also possible to have reflection over the around the point (1, 1) maps the parallelogram onto diagonal lines. itself.

9. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure ANSWER: does not have any lines of of symmetry. yes; 4 ANSWER: no

10. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line. 11. The given figure has reflectional symmetry. SOLUTION: In order for the figure to map onto itself, the line of A figure has reflectional symmetry if the figure can reflection must go through the center point. be mapped onto itself by a reflection in a line.

The figure has a vertical and horizontal line of The given hexagon has reflectional symmetry. reflection. In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though opposites edges.

It is also possible to have reflection over the diagonal lines.

There are three lines of reflection that go though opposites vertices.

Therefore, the figure has four lines of symmetry

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry.

ANSWER: yes; 4

ANSWER: yes; 6

11. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry. 12.

SOLUTION: In order for the hexagon to map onto itself, the line of reflection must go through the center point. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

There are three lines of reflection that go though The figure has reflectional symmetry. opposites edges.

There is only one line of symmetry, a horizontal line through the middle of the figure.

There are three lines of reflection that go though Thus, the figure has one line of symmetry. opposites vertices. ANSWER: yes; 1

The figure has reflectional symmetry.

There is only one possible line of reflection, horizontally though the middle of the figure. ANSWER: yes; 6

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

12. SOLUTION: 14. A figure has reflectional symmetry if the figure can SOLUTION: be mapped onto itself by a reflection in a line. A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

The given figure does not have reflectional There is only one line of symmetry, a horizontal line symmetry. It is not possible to draw a line of through the middle of the figure. reflection where the figure can map onto itself.

ANSWER: no

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same 13. location above and below the center horizontal line, the flag would have three lines of symmetry. SOLUTION: A figure has reflectional symmetry if the figure can ANSWER: be mapped onto itself by a reflection in a line. no

The figure has reflectional symmetry. 16. Refer to the flag on page 262.

There is only one possible line of reflection, SOLUTION: horizontally though the middle of the figure. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

Thus, the figure has one line of symmetry. In order for the figure to map onto itself, the line of

ANSWER: reflection must go through the center point.

yes; 1 A horizontal and vertical lines of reflection are possible.

14. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. Two diagonal lines of reflection are possible. The given figure does not have reflectional symmetry. It is not possible to draw a line of reflection where the figure can map onto itself.

ANSWER: no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their number. 15. Refer to page 262. There are a total of four possible lines that go through the center and are lines of reflections. Thus, SOLUTION: the flag has four lines of symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER: no

16. Refer to the flag on page 262. ANSWER: yes; 4 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point. 3-5 Symmetry A horizontal and vertical lines of reflection are possible. 17. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

Two diagonal lines of reflection are possible.

It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry.

ANSWER:

yes; 1 There are a total of four possible lines that go through the center and are lines of reflections. Thus, the flag has four lines of symmetry.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

ANSWER: 18. yes; 4 SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry. 17. Refer to page 262. The number of times a figure maps onto itself as it SOLUTION: rotates from 0° to 360° is called the order of eSolutionsA figureManual has- Powered reflectionalby Cognero symmetry if the figure can symmetry. Page 7 be mapped onto itself by a reflection in a line. This figure has order 2 rotational symmetry, since The figure has reflectional symmetry. you have to rotate 180° to get the figure to map onto itself. A horizontal line is a line of reflections for this flag. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has a magnitude of symmetry of . It is not possible to reflect over a vertical or line through the diagonals. ANSWER:

Thus, the figure has one line of symmetry.

ANSWER: yes; 1 yes; 2; 180°

19. State whether the figure has rotational SOLUTION: symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state A figure in the plane has rotational symmetry if the the order and magnitude of symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The triangle has rotational symmetry. 18. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has rotational symmetry. The figure has order 3 rotational symmetry.

This figure has order 2 rotational symmetry, since The figure has magnitude of symmetry of you have to rotate 180° to get the figure to map onto . itself. ANSWER: The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has a magnitude of symmetry of . yes; 3; 120°

ANSWER:

20.

SOLUTION: yes; 2; 180° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be 19. mapped onto itself. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation ANSWER: between 0° and 360° about the center of the figure. no

The triangle has rotational symmetry.

21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 3; 120°

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation ANSWER: between 0° and 360° about the center of the figure. no The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

22. ANSWER: no SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it

can be mapped onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 8; 45°

ANSWER: no 23. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 22. between 0° and 360° about the center of the figure.

SOLUTION: The figure has rotational symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This means that the figure can be rotated 8 times and map onto itself within 360°. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The magnitude of symmetry is the smallest angle symmetry. through which a figure can be rotated so that it maps onto itself. The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it The figure has magnitude of symmetry of map onto itself within 360°. .

The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of . yes; 8; 45° ANSWER: WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, state the order and magnitude of symmetry. 24. Refer to page 263. SOLUTION: A figure in the plane has rotational symmetry if the yes; 8; 45° figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The wheel has rotational symmetry.

23. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of SOLUTION: symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The wheel has order 5 rotational symmetry. There between 0° and 360° about the center of the figure. are 5 large spokes and 5 small spokes. You can rotate the wheel 5 times within 360° and map the The figure has rotational symmetry. figure onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The wheel has magnitude of symmetry

The number of times a figure maps onto itself as it . rotates from 0° to 360° is called the order of ANSWER: symmetry. yes; 5; 72° The figure has order 8 rotational symmetry. This means that the figure can be rotated 8 times and map 25. Refer to page 263. onto itself within 360°. SOLUTION: The magnitude of symmetry is the smallest angle A figure in the plane has rotational symmetry if the through which a figure can be rotated so that it maps figure can be mapped onto itself by a rotation onto itself. between 0° and 360° about the center of the figure.

The figure has magnitude of symmetry of The wheel has rotational symmetry. . The number of times a figure maps onto itself as it ANSWER: rotates from 0° to 360° is called the order of symmetry.

The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself. yes; 8; 45° The magnitude of symmetry is the smallest angle WHEELS State whether each wheel cover appears through which a figure can be rotated so that it maps to have rotational symmetry. Write yes or no. If so, onto itself. state the order and magnitude of symmetry. 24. Refer to page 263. The wheel has order 8 rotational symmetry and SOLUTION: magnitude . A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation ANSWER: between 0° and 360° about the center of the figure. yes; 8; 45°

26. Refer to page 263. The wheel has rotational symmetry. SOLUTION: The number of times a figure maps onto itself as it A figure in the plane has rotational symmetry if the rotates from 0° to 360° is called the order of figure can be mapped onto itself by a rotation symmetry. between 0° and 360° about the center of the figure. The wheel has rotational symmetry. The wheel has order 5 rotational symmetry. There are 5 large spokes and 5 small spokes. You can The number of times a figure maps onto itself as it rotate the wheel 5 times within 360° and map the rotates from 0° to 360° is called the order of figure onto itself. symmetry. The wheel has order 10 rotational symmetry. There are 10 bolts and the tire can be The magnitude of symmetry is the smallest angle rotated 10 times within 360° and map onto itself. through which a figure can be rotated so that it maps onto itself. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The wheel has magnitude of symmetry onto itself. The wheel has magnitude of symmetry of . .

ANSWER: ANSWER: yes; 5; 72° yes; 10; 36° State whether the figure has line symmetry 25. Refer to page 263. and/or rotational symmetry. If so, describe the SOLUTION: reflections and/or rotations that map the figure A figure in the plane has rotational symmetry if the onto itself. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The wheel has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of

symmetry. 27. SOLUTION: The wheel has order 8 rotational symmetry. There This triangle is scalene, so it cannot have symmetry. are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself. ANSWER: no symmetry The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The wheel has order 8 rotational symmetry and magnitude .

ANSWER: 28. yes; 8; 45° SOLUTION: 26. Refer to page 263. This figure is a square, because each pair of adjacent SOLUTION: sides is congruent and perpendicular. All squares have both line and rotational symmetry. A figure in the plane has rotational symmetry if the The line symmetry is vertically, horizontally, and figure can be mapped onto itself by a rotation diagonally through the center of the square, with lines between 0° and 360° about the center of the figure. that are either parallel to the sides of the square or The wheel has rotational symmetry. that include two vertices of the square. The equations of those lines are: x = 0, y = 0, y = x, and y The number of times a figure maps onto itself as it = -x rotates from 0° to 360° is called the order of symmetry. The wheel has order 10 rotational The rotational symmetry is for each quarter turn in a symmetry. There are 10 bolts and the tire can be square, so the rotations of 90, 180, and 270 degrees rotated 10 times within 360° and map onto itself. around the origin map the square onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps ANSWER: onto itself. The wheel has magnitude of symmetry of line symmetry; rotational symmetry; the reflection in . the line x = 0, the reflection in the line y = 0, the reflection in the line y = x, and the reflection in the ANSWER: line y = -x all map the square onto itself; the rotations yes; 10; 36° of 90, 180, and 270 degrees around the origin map the square onto itself. State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

29. SOLUTION: The trapezoid has line symmetry, because it is 27. isosceles, but it does not have rotational symmetry, because no trapezoid does. SOLUTION: This triangle is scalene, so it cannot have symmetry. The reflection in the line y = 1.5 maps the trapezoid onto itself, because that is the perpendicular bisector ANSWER: to the parallel sides. no symmetry ANSWER: line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself.

28. SOLUTION: This figure is a square, because each pair of adjacent 30. sides is congruent and perpendicular. All squares have both line and rotational symmetry. SOLUTION: The line symmetry is vertically, horizontally, and This figure is a parallelogram, so it has rotational diagonally through the center of the square, with lines symmetry of a half turn or 180 degrees around its that are either parallel to the sides of the square or center, which is the point (1, -1.5). that include two vertices of the square. The equations of those lines are: x = 0, y = 0, y = x, and y Since this parallelogram is not a rhombus it does not = -x have line symmetry.

ANSWER: 31. MODELING Symmetry is an important component line symmetry; rotational symmetry; the reflection in of photography. Photographers often use reflection in the line x = 0, the reflection in the line y = 0, the water to create symmetry in photos. The photo on reflection in the line y = x, and the reflection in the page 263 is a long exposure shot of the Eiffel tower line y = -x all map the square onto itself; the rotations reflected in a pool. of 90, 180, and 270 degrees around the origin map the square onto itself. a. Describe the two-dimensional symmetry created by the photo. b. Is there rotational symmetry in the photo? Explain your reasoning. SOLUTION: a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. 29. There is a vertical line of symmetry through the SOLUTION: center of the photo. The trapezoid has line symmetry, because it is b No; sample answer: Because of how the image is isosceles, but it does not have rotational symmetry, reflected over the horizontal line, there is no because no trapezoid does. rotational symmetry. ANSWER: The reflection in the line y = 1.5 maps the trapezoid onto itself, because that is the perpendicular bisector a. Sample answer: There is a horizontal line of to the parallel sides. symmetry between the tower and its reflection. There is a vertical line of symmetry through the ANSWER: center of the photo. line symmetry; the reflection in the line y = 1.5 maps b No; sample answer: Because of how the image is the trapezoid onto itself. reflected over the horizontal line, there is no rotational symmetry.

COORDINATE GEOMETRY Determine whether the figure with the given vertices has line symmetry and/or rotational symmetry. 32. R(–3, 3), S(–3, –3), T(3, 3) SOLUTION: 30. Draw the figure on a coordinate plane. SOLUTION: This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its center, which is the point (1, -1.5).

Since this parallelogram is not a rhombus it does not have line symmetry.

ANSWER: rotational symmetry; the rotation of 180 degrees A figure has line symmetry if the figure can be around the point (1, -1.5) maps the parallelogram mapped onto itself by a reflection in a line. onto itself. The given triangle has a line of symmetry through 31. MODELING Symmetry is an important component points (0, 0) and (–3, 3). of photography. Photographers often use reflection in water to create symmetry in photos. The photo on A figure in the plane has rotational symmetry if the page 263 is a long exposure shot of the Eiffel tower figure can be mapped onto itself by a rotation reflected in a pool. between 0° and 360° about the center of the figure. There is not way to rotate the figure and have it map a. Describe the two-dimensional symmetry created onto itself. by the photo. b. Is there rotational symmetry in the photo? Explain Thus, the figure has only line symmetry. your reasoning. ANSWER: SOLUTION: line a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4)

center of the photo. SOLUTION: b No; sample answer: Because of how the image is Draw the figure on a coordinate plane. reflected over the horizontal line, there is no rotational symmetry.

ANSWER: a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no rotational symmetry. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The COORDINATE GEOMETRY Determine given figure has 4 lines of symmetry. The line of whether the figure with the given vertices has line symmetry are though the following pairs of points symmetry and/or rotational symmetry. {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, 32. R(–3, 3), S(–3, –3), T(3, 3) and {(2, 2), (2, –2)}.

SOLUTION: A figure in the plane has rotational symmetry if the Draw the figure on a coordinate plane. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 2) The given triangle has a line of symmetry through SOLUTION: points (0, 0) and (–3, 3). Draw the figure on a coordinate plane.

Thus, the figure has only line symmetry.

ANSWER: A figure has line symmetry if the figure can be line mapped onto itself by a reflection in a line. The given hexagon has 2 lines of symmetry. The lines pass through the following pair of points {(0, 4), (0, – 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) 4)}, and {(3, 0), (–3, 0)} SOLUTION: Draw the figure on a coordinate plane. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure has rotational symmetry. You can rotate the figure once within 360° and have it map to itself.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER:

A figure has line symmetry if the figure can be line and rotational mapped onto itself by a reflection in a line. The 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) given figure has 4 lines of symmetry. The line of symmetry are though the following pairs of points SOLUTION: {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, Draw the figure on a coordinate plane. and {(2, 2), (2, –2)}.

Thus, the figure has both symmetry and line rotational symmetry. A figure has line symmetry if the figure can be ANSWER: mapped onto itself by a reflection in a line. The trapezoid has a line of reflection through points (0,3) line and rotational and (0, –3). 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 2) A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the Draw the figure on a coordinate plane. figure. There is no way to rotate this figure and have it map onto itself. Thus, it does not have rotational symmetry.

Therefore, the figure has only line symmetry.

ANSWER: line

A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation Graph the function. between 0° and 360° about the center of the figure. The figure has rotational symmetry. You can rotate the figure once within 360° and have it map to itself.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational

35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) A figure has reflectional symmetry if the figure can SOLUTION: be mapped onto itself by a reflection in a line. The Draw the figure on a coordinate plane. line y = x has reflectional symmetry since any line perpendicular to y = x is a line of reflection. The equation of the line symmetry is y = –x.

The magnitude of symmetry is the smallest angle A figure in the plane has rotational symmetry if the through which a figure can be rotated so that it maps figure can be mapped onto itself by a rotation onto itself. between 0° and 360° about the center of the The graph has magnitude of symmetry of figure. There is no way to rotate this figure and have . it map onto itself. Thus, it does not have rotational symmetry. Thus, the graph has both reflectional and rotational Therefore, the figure has only line symmetry. symmetry.

ANSWER: ANSWER: line rotational; 2; 180°; line symmetry; y = –x

2 37. y = x + 1 SOLUTION: Graph the function.

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The line y = x has reflectional symmetry since any line

perpendicular to y = x is a line of reflection. The equation of the line symmetry is y = –x. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The A figure in the plane has rotational symmetry if the graph is reflected through the y-axis. Thus, the figure can be mapped onto itself by a rotation equation of the line symmetry is x = 0. between 0° and 360° about the center of the figure.

The line can be rotated twice within 360° and be mapped onto itself. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the The number of times a figure maps onto itself as it figure. There is no way to rotate the graph and have rotates from 0° to 360° is called the order of it map onto itself. symmetry. The graph has order 2 rotational

symmetry. Thus, the graph has only reflectional symmetry. The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps line; x = 0 onto itself. The graph has magnitude of symmetry of .

Thus, the graph has both reflectional and rotational symmetry.

ANSWER: rotational; 2; 180°; line symmetry; y = –x

38. y = –x3 SOLUTION: Graph the function.

2 37. y = x + 1 SOLUTION: Graph the function.

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. You can rotate the graph through the origin

A figure has reflectional symmetry if the figure can and have it map onto itself. be mapped onto itself by a reflection in a line. The graph is reflected through the y-axis. Thus, the The number of times a figure maps onto itself as it equation of the line symmetry is x = 0. rotates from 0° to 360° is called the order of The graph has order 2 rotational symmetry. symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the The magnitude of symmetry is the smallest angle figure. There is no way to rotate the graph and have through which a figure can be rotated so that it maps it map onto itself. onto itself. The graph has magnitude of symmetry of . Thus, the graph has only reflectional symmetry. Thus, the graph has only rotational symmetry. ANSWER: line; x = 0 ANSWER: rotational; 2; 180°

38. y = –x3 39. Refer to the rectangle on the coordinate plane. SOLUTION: Graph the function.

a. What are the equations of the lines of symmetry of the rectangle? b. What happens to the equations of the lines of symmetry when the rectangle is rotated 90 degrees A figure has reflectional symmetry if the figure can counterclockwise around its center of symmetry? be mapped onto itself by a reflection in a line. The Explain. graph does not have a line of reflections where the graph can be mapped onto itself. SOLUTION: a. The lines of symmetry are parallel to the sides of A figure in the plane has rotational symmetry if the the rectangles, and through the center of rotation. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the The slopes of the sides of the rectangle are 0.5 and figure. You can rotate the graph through the origin and have it map onto itself. -2, so the slopes of the lines of symmetry are the same. The number of times a figure maps onto itself as it The center of the rectangle is (1, 1.5). Use the rotates from 0° to 360° is called the order of point-slope formula to find equations. The graph has order 2 rotational symmetry. symmetry.

Thus, the graph has only rotational symmetry. ANSWER: a. ANSWER: b. The equations of the lines of symmetry do not rotational; 2; 180° change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to 39. Refer to the rectangle on the coordinate plane. draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a

square, regular pentagon, and regular hexagon. a. What are the equations of the lines of symmetry of Record the number of lines of symmetry and the the rectangle? order of symmetry for each polygon. b. What happens to the equations of the lines of d. Verbal Make a conjecture about the number of symmetry when the rectangle is rotated 90 degrees lines of symmetry and the order of symmetry for a counterclockwise around its center of symmetry? regular polygon with n sides. Explain. SOLUTION: SOLUTION: a. Construct an equilateral triangle and label the a. The lines of symmetry are parallel to the sides of vertices A, B, and C. Draw a line through A the rectangles, and through the center of rotation. perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image The slopes of the sides of the rectangle are 0.5 and maps to the original, then this line is a line of -2, so the slopes of the lines of symmetry are the reflection. same. The center of the rectangle is (1, 1.5). Use the point-slope formula to find equations.

b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other. Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the ANSWER: reflected image. If the image maps to the original, a. then this line is a line of reflection. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

perpendicular to . Reflect the triangle in the line. There are 3 lines of symmetry. Show the labels of the reflected image. If the image maps to the original, then this line is a line of b. Construct an equilateral triangle and show the reflection. labels of the vertices. Next, find the center of the triangle. Since this is an equilateral triangle, the circumcenter, incenter, centroid, and orthocenter are the same point. Construct altitudes through each vertex and label the intersection. Rotate the triangle about point D. A 120 degree rotation will map the image to the original. Show the labels of the image.

Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

There are 3 lines of symmetry.

vertex and label the intersection. Since the figure maps onto itself 3 times as it is Rotate the triangle about point D. A 120 degree rotated, the order of symmetry is 3. rotation will map the image to the original. Show the c. labels of the image. Square Construct a square and then construct lines through the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. The triangle can be rotated a third time about D. A Next, rotate the square about the center point. The 360 degree rotation maps the image to the original. image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5.

Since the figure maps onto itself 3 times as it is rotated, the order of symmetry is 3. c. Square Regular Hexagon Construct a square and then construct lines through Construct a regular hexagon and then construct lines the midpoints of each side and diagonals. Use the through each vertex perpendicular to the sides. Use reflection tool first to find that the image maps onto the reflection tool first to find that the image maps the original when reflected in each of the 4 lines onto the original when reflected in each of the 6 lines constructed. So there are 4 lines of symmetry. constructed. So there are 6 lines of symmetry. Next, rotate the square about the center point. The Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 image maps to the original at 60, 120, 180, 240, 300, degree rotations. So the order of symmetry is 4. and 360 degree rotations. So the order of symmetry is 6.

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use

the reflection tool first to find that the image maps d. Sample answer: for each figure studied, the onto the original when reflected in each of the 5 lines number of sides of the figure is the same as the lines constructed. So there are 5 lines of symmetry. of symmetry and the order of symmetry. A regular Next, rotate the square about the center point. The polygon with n sides has n lines of symmetry and image maps to the original at 72, 144, 216, 288, and order of symmetry n. 360 degree rotations. So the order of symmetry is 5. ANSWER: a. 3 b. 3 c.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

is 6. figure has 4 lines of symmetry.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure also has rotational symmetry. d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines Therefore, neither of them are correct. Figure A has of symmetry and the order of symmetry. A regular both line and rotational symmetry. polygon with n sides has n lines of symmetry and order of symmetry n. ANSWER: Neither; Figure A has both line and rotational ANSWER: symmetry. a. 3 b. 3 42. CHALLENGE A quadrilateral in the coordinate c. plane has exactly two lines of symmetry, y = x – 1 and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of symmetry. d. Sample answer: A regular polygon with n sides SOLUTION: has n lines of symmetry and order of symmetry n. Graph the figure and the lines of symmetry. 41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

and b = .

A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2).

A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) between 0° and 360° about the center of the figure.

The figure also has rotational symmetry.

Therefore, neither of them are correct. Figure A has both line and rotational symmetry. 43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. ANSWER: SOLUTION: Neither; Figure A has both line and rotational symmetry. circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such 42. CHALLENGE A quadrilateral in the coordinate lines. plane has exactly two lines of symmetry, y = x – 1 ANSWER: and y = –x + 2. Find a set of possible vertices for circle; Every line through the center of a circle is a the figure. Graph the figure and the lines of line of symmetry, and there are infinitely many such symmetry. lines. SOLUTION: 44. OPEN-ENDED Draw a figure with line symmetry Graph the figure and the lines of symmetry. but not rotational symmetry. Explain. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not Pick points that are the same distance a from one have rotational symmetry. line and the same distance b from the other line. In the same answer, the quadrilateral is a rectangle with An isosceles triangle has line symmetry from the sides which are parallel to the lines of symmetry. vertex angle to the base of the triangle, but it does This guarantees that the vertices of the quadrilateral not have rotational symmetry because it cannot be are the same distance a from one line and the same rotated from 0° to 360° and map onto itself. distance b from the other line. In this case, a =

and b = .

A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2). ANSWER: ANSWER: Sample answer: An isosceles triangle has line symmetry from the vertex angle to the base of the Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

45. WRITING IN MATH How are line symmetry and rotational symmetry related? 43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. SOLUTION: In both types of symmetries the figure is mapped SOLUTION: onto itself. circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such Rotational symmetry. lines.

ANSWER: circle; Every line through the center of a circle is a Reflectional symmetry: line of symmetry, and there are infinitely many such lines.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. SOLUTION: A figure has line symmetry if the figure can be In some cases an object can have both rotational and mapped onto itself by a reflection in a line. A figure reflectional symmetry, such as the diamond, however in the plane has rotational symmetry if the figure can some objects do not have both such as the crab. be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not have rotational symmetry.

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be ANSWER: rotated from 0° to 360° and map onto itself. Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry. ANSWER: 46. Sasha owns a tile store. For each tile in her store, she Sample answer: An isosceles triangle has line calculates the sum of the number of lines of symmetry from the vertex angle to the base of the symmetry and the order of symmetry, and then she triangle, but it does not have rotational symmetry enters this value into a database. Which value should because it cannot be rotated from 0° to 360° and she enter in the database for the tile shown here? map onto itself.

45. WRITING IN MATH How are line symmetry and A 2 rotational symmetry related? B 3 SOLUTION: C 4 In both types of symmetries the figure is mapped D 8 onto itself. SOLUTION:

Rotational symmetry.

Reflectional symmetry: The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn around its center.

In some cases an object can have both rotational and 2 + 2 = 4, so C is the correct answer. reflectional symmetry, such as the diamond, however some objects do not have both such as the crab. ANSWER:

47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A

ANSWER: B Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line symmetry the figure is mapped onto itself by a

reflection, and in rotational symmetry, a figure is C mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry.

46. Sasha owns a tile store. For each tile in her store, she D calculates the sum of the number of lines of symmetry and the order of symmetry, and then she enters this value into a database. Which value should E she enter in the database for the tile shown here?

SOLUTION: Option A has rotational and reflectional symmetry.

A 2 B 3 C 4 Option B has reflectional symmetry but not rotational D 8 symmetry. SOLUTION:

Option C has neither rotational nor reflectional symmetry. The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn Option D has rotational symmetry but not reflectional around its center. symmetry.

2 + 2 = 4, so C is the correct answer.

The correct choice is D.

C ANSWER: D

48. Which of the following figures may have exactly one

D line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle E C Isosceles triangle D Scalene triangle SOLUTION: SOLUTION: An isosceles triangle has one line of symmetry and Option A has rotational and reflectional symmetry. no rotational symmetry. The correct choice is C.

ANSWER: C

Option B has reflectional symmetry but not rotational 49. Camryn plotted the points , symmetry. and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C Option C has neither rotational nor reflectional D symmetry. SOLUTION: First, plot the points.

Option D has rotational symmetry but not reflectional symmetry.

Option E has reflectional symmetry but not rotational symmetry.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

The correct choice is D.

ANSWER: D

ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D SOLUTION: First, plot the points.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

Thus, there are four possible lines that go through the center and are lines of reflections.

State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

Therefore, the figure has four lines of symmetry.

ANSWER:

1. yes; 4 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure. 2. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto Two lines of reflection go through the vertices of the itself. figure. ANSWER: no

Thus, there are four possible lines that go through SOLUTION: the center and are lines of reflections. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

The figure has a vertical line of symmetry.

Therefore, the figure has four lines of symmetry.

ANSWER: yes; 4 It does not have a horizontal line of symmetry.

The figure does not have a line of symmetry through the vertices.

SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. Thus, the figure has only one line of symmetry.

The given figure does not have reflectional ANSWER: symmetry. There is no way to fold or reflect it onto yes; 1 itself.

ANSWER: no

The figure does not have a line of symmetry through the vertices.

5. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Thus, the figure has only one line of symmetry. The given figure has rotational symmetry.

ANSWER: yes; 1

The number of times a figure maps onto itself as it State whether the figure has rotational symmetry. Write yes or no. If so, copy the rotates form 0° and 360° is called the order of figure, locate the center of symmetry, and state symmetry. the order and magnitude of symmetry. The given figure has order of symmetry of 2, since the figure can be rotated twice in 360°.

The magnitude of symmetry is the smallest angle 4. through which a figure can be rotated so that it maps

SOLUTION: onto itself.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation Since the figure has order 2 rotational symmetry, the between 0° and 360° about the center of the figure. magnitude of the symmetry is .

ANSWER: For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure yes; 2; 180° were a regular pentagon, it would have rotational symmetry.

ANSWER: no

between 0° and 360° about the center of the figure. between 0° and 360° about the center of the figure.

The given figure has rotational symmetry. The given figure has rotational symmetry.

The number of times a figure maps onto itself as it

rotates from 0° to 360° is called the order of symmetry. The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of Since the figure can be rotated 4 times within 360° , symmetry. it has order 4 rotational symmetry

Since the figure has order 2 rotational symmetry, the ANSWER: magnitude of the symmetry is . yes; 4; 90°

ANSWER: yes; 2; 180°

State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

6. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. 7.

SOLUTION: The given figure has rotational symmetry. Vertical and horizontal lines through the center and diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way.

The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1.

The figure has magnitude of symmetry of .

ANSWER: yes; 4; 90°

8. SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent.

REGULARITY State whether the figure appears to have line symmetry. Write yes or no. 7. If so, copy the figure, draw all lines of symmetry, and state their number. SOLUTION: Vertical and horizontal lines through the center and diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way. 9. The equations of those lines in this figure are x = 0, SOLUTION: y = -1, y = x - 1, and y = -x - 1. A figure has reflectional symmetry if the figure can Each quarter turn also maps the square onto itself. be mapped onto itself by a reflection in a line. So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself. For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure ANSWER: does not have any lines of of symmetry. line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the ANSWER: reflection in the line y = x - 1, and the reflection in no the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

10. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

8. The given figure has reflectional symmetry.

SOLUTION: In order for the figure to map onto itself, the line of This figure does not have line symmetry, because reflection must go through the center point. adjacent sides are not congruent.

ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

9. Therefore, SOLUTION: the figure has four lines of symmetry A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry.

ANSWER: no ANSWER: yes; 4

10. SOLUTION: A figure has reflectional line symmetry if the figure

can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point. 11.

The figure has a vertical and horizontal line of SOLUTION: reflection. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point.

It is also possible to have reflection over the There are three lines of reflection that go though diagonal lines. opposites edges.

Therefore, the figure has four lines of symmetry There are three lines of reflection that go though opposites vertices.

ANSWER: There are six possible lines that go through the center yes; 4 and are lines of reflections. Thus, the hexagon has six lines of symmetry.

ANSWER: yes; 6 11. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point. 12. There are three lines of reflection that go though SOLUTION: opposites edges. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

There is only one line of symmetry, a horizontal line through the middle of the figure. There are three lines of reflection that go though opposites vertices.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry.

13. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

ANSWER: The figure has reflectional symmetry. yes; 6 There is only one possible line of reflection, horizontally though the middle of the figure.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

12. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. 14.

SOLUTION: There is only one line of symmetry, a horizontal line A figure has reflectional symmetry if the figure can through the middle of the figure. be mapped onto itself by a reflection in a line.

SOLUTION: The flag does not have any reflectional symmetry. If A figure has reflectional symmetry if the figure can the red lines in the diagonals were in the same be mapped onto itself by a reflection in a line. location above and below the center horizontal line, the flag would have three lines of symmetry. The figure has reflectional symmetry. ANSWER: There is only one possible line of reflection, no horizontally though the middle of the figure. 16. Refer to the flag on page 262. SOLUTION: A figure has reflectional symmetry if the figure can Thus, the figure has one line of symmetry. be mapped onto itself by a reflection in a line. ANSWER:

yes; 1 The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

14. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. It is not possible to draw a line of reflection where the figure can map onto itself. Two diagonal lines of reflection are possible. ANSWER: no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their number. 15. Refer to page 262.

SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. There are a total of four possible lines that go through the center and are lines of reflections. Thus, The flag does not have any reflectional symmetry. If the flag has four lines of symmetry. the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER: no

16. Refer to the flag on page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. ANSWER: yes; 4 The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

17. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

Two diagonal lines of reflection are possible. The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

It is not possible to reflect over a vertical or line through the diagonals.

There are a total of four possible lines that go through the center and are lines of reflections. Thus, Thus, the figure has one line of symmetry. the flag has four lines of symmetry. ANSWER: yes; 1

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state ANSWER: the order and magnitude of symmetry. yes; 4

18. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

17. Refer to page 262. SOLUTION:

A figure has reflectional symmetry if the figure can The figure has rotational symmetry.

be mapped onto itself by a reflection in a line.

The number of times a figure maps onto itself as it

The figure has reflectional symmetry. rotates from 0° to 360° is called the order of symmetry. A horizontal line is a line of reflections for this flag. This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto itself.

The magnitude of symmetry is the smallest angle

It is not possible to reflect over a vertical or line through which a figure can be rotated so that it maps onto itself. through the diagonals.

The figure has a magnitude of symmetry of Thus, the figure has one line of symmetry. .

ANSWER: ANSWER: yes; 1

3-5 Symmetry yes; 2; 180°

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. 19. SOLUTION: A figure in the plane has rotational symmetry if the 18. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. SOLUTION: A figure in the plane has rotational symmetry if the The triangle has rotational symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

This figure has order 2 rotational symmetry, since The magnitude of symmetry is the smallest angle you have to rotate 180° to get the figure to map onto through which a figure can be rotated so that it maps itself. onto itself.

The magnitude of symmetry is the smallest angle The figure has magnitude of symmetry of through which a figure can be rotated so that it maps . onto itself. ANSWER: The figure has a magnitude of symmetry of .

ANSWER:

yes; 3; 120°

yes; 2; 180° 20. SOLUTION: A figure in the plane has rotational symmetry if the 19. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. SOLUTION: A figure in the plane has rotational symmetry if the The isosceles trapezoid has no rotational symmetry. figure can be mapped onto itself by a rotation eSolutions Manual - Powered by Cognero There is no way to rotate it such that it can be Page 8 between 0° and 360° about the center of the figure. mapped onto itself.

The triangle has rotational symmetry. ANSWER: no

21.

SOLUTION: The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of A figure in the plane has rotational symmetry if the symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure has order 3 rotational symmetry. The crescent shaped figure has no rotational The magnitude of symmetry is the smallest angle symmetry. There is no way to rotate it such that it

through which a figure can be rotated so that it maps can be mapped onto itself. onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 3; 120°

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself. ANSWER: no

ANSWER: no

22. SOLUTION: A figure in the plane has rotational symmetry if the 21. figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The figure has rotational symmetry. between 0° and 360° about the center of the figure.

The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

ANSWER: no yes; 8; 45°

23. 22. SOLUTION: SOLUTION: A figure in the plane has rotational symmetry if the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. between 0° and 360° about the center of the figure.

The figure has rotational symmetry. The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The figure has order 8 rotational symmetry. This symmetry. means that the figure can be rotated 8 times and map onto itself within 360°. The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it The magnitude of symmetry is the smallest angle map onto itself within 360°. through which a figure can be rotated so that it maps onto itself. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The figure has magnitude of symmetry of onto itself. .

The figure has magnitude of symmetry of ANSWER: .

ANSWER:

yes; 8; 45°

WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so,

yes; 8; 45° state the order and magnitude of symmetry. 24. Refer to page 263. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 23. between 0° and 360° about the center of the figure.

SOLUTION: The wheel has rotational symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The number of times a figure maps onto itself as it between 0° and 360° about the center of the figure. rotates from 0° to 360° is called the order of symmetry. The figure has rotational symmetry. The wheel has order 5 rotational symmetry. There are 5 large spokes and 5 small spokes. You can rotate the wheel 5 times within 360° and map the figure onto itself.

24. Refer to page 263. The magnitude of symmetry is the smallest angle SOLUTION: through which a figure can be rotated so that it maps A figure in the plane has rotational symmetry if the onto itself. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The wheel has order 8 rotational symmetry and magnitude . The wheel has rotational symmetry. ANSWER: The number of times a figure maps onto itself as it yes; 8; 45° rotates from 0° to 360° is called the order of symmetry. 26. Refer to page 263. SOLUTION: The wheel has order 5 rotational symmetry. There A figure in the plane has rotational symmetry if the are 5 large spokes and 5 small spokes. You can figure can be mapped onto itself by a rotation rotate the wheel 5 times within 360° and map the between 0° and 360° about the center of the figure. figure onto itself. The wheel has rotational symmetry.

The magnitude of symmetry is the smallest angle The number of times a figure maps onto itself as it through which a figure can be rotated so that it maps rotates from 0° to 360° is called the order of onto itself. symmetry. The wheel has order 10 rotational symmetry. There are 10 bolts and the tire can be The wheel has magnitude of symmetry rotated 10 times within 360° and map onto itself. . The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps yes; 5; 72° onto itself. The wheel has magnitude of symmetry of . 25. Refer to page 263. ANSWER: SOLUTION: yes; 10; 36° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation State whether the figure has line symmetry between 0° and 360° about the center of the figure. and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure The wheel has rotational symmetry. onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself. 27. SOLUTION: The magnitude of symmetry is the smallest angle This triangle is scalene, so it cannot have symmetry. through which a figure can be rotated so that it maps onto itself. ANSWER: no symmetry The wheel has order 8 rotational symmetry and magnitude .

ANSWER: yes; 8; 45° 26. Refer to page 263. SOLUTION: 28. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. This figure is a square, because each pair of adjacent The wheel has rotational symmetry. sides is congruent and perpendicular. All squares have both line and rotational symmetry. The number of times a figure maps onto itself as it The line symmetry is vertically, horizontally, and rotates from 0° to 360° is called the order of diagonally through the center of the square, with lines symmetry. The wheel has order 10 rotational that are either parallel to the sides of the square or symmetry. There are 10 bolts and the tire can be that include two vertices of the square. The rotated 10 times within 360° and map onto itself. equations of those lines are: x = 0, y = 0, y = x, and y = -x

The magnitude of symmetry is the smallest angle The rotational symmetry is for each quarter turn in a through which a figure can be rotated so that it maps square, so the rotations of 90, 180, and 270 degrees onto itself. The wheel has magnitude of symmetry of around the origin map the square onto itself. .

the square onto itself.

The reflection in the line y = 1.5 maps the trapezoid onto itself, because that is the perpendicular bisector to the parallel sides.

ANSWER: line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself. 28. SOLUTION: This figure is a square, because each pair of adjacent sides is congruent and perpendicular. All squares have both line and rotational symmetry. The line symmetry is vertically, horizontally, and diagonally through the center of the square, with lines that are either parallel to the sides of the square or 30. that include two vertices of the square. The SOLUTION: equations of those lines are: x = 0, y = 0, y = x, and y This figure is a parallelogram, so it has rotational = -x symmetry of a half turn or 180 degrees around its

center, which is the point (1, -1.5). The rotational symmetry is for each quarter turn in a

square, so the rotations of 90, 180, and 270 degrees Since this parallelogram is not a rhombus it does not around the origin map the square onto itself. have line symmetry.

ANSWER: ANSWER: rotational symmetry; the rotation of 180 degrees line symmetry; rotational symmetry; the reflection in around the point (1, -1.5) maps the parallelogram the line x = 0, the reflection in the line y = 0, the onto itself. reflection in the line y = x, and the reflection in the line y = -x all map the square onto itself; the rotations 31. MODELING Symmetry is an important component of 90, 180, and 270 degrees around the origin map of photography. Photographers often use reflection in

the square onto itself. water to create symmetry in photos. The photo on page 263 is a long exposure shot of the Eiffel tower reflected in a pool.

COORDINATE GEOMETRY Determine 30. whether the figure with the given vertices has line symmetry and/or rotational symmetry. SOLUTION: 32. R(–3, 3), S(–3, –3), T(3, 3) This figure is a parallelogram, so it has rotational SOLUTION: symmetry of a half turn or 180 degrees around its center, which is the point (1, -1.5). Draw the figure on a coordinate plane.

Since this parallelogram is not a rhombus it does not have line symmetry.

ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram onto itself.

a. Describe the two-dimensional symmetry created A figure in the plane has rotational symmetry if the by the photo. figure can be mapped onto itself by a rotation b. Is there rotational symmetry in the photo? Explain between 0° and 360° about the center of the figure.

your reasoning. There is not way to rotate the figure and have it map SOLUTION: onto itself. a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. Thus, the figure has only line symmetry. There is a vertical line of symmetry through the ANSWER: center of the photo. b No; sample answer: Because of how the image is line reflected over the horizontal line, there is no rotational symmetry. 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) ANSWER: SOLUTION: a. Sample answer: There is a horizontal line of Draw the figure on a coordinate plane. symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no rotational symmetry.

COORDINATE GEOMETRY Determine whether the figure with the given vertices has line symmetry and/or rotational symmetry. A figure has line symmetry if the figure can be 32. R(–3, 3), S(–3, –3), T(3, 3) mapped onto itself by a reflection in a line. The SOLUTION: given figure has 4 lines of symmetry. The line of symmetry are though the following pairs of points Draw the figure on a coordinate plane. {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, and {(2, 2), (2, –2)}.

Thus, the figure has both line symmetry and A figure has line symmetry if the figure can be rotational symmetry. mapped onto itself by a reflection in a line. ANSWER: The given triangle has a line of symmetry through line and rotational points (0, 0) and (–3, 3). 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – A figure in the plane has rotational symmetry if the 2) figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. Draw the figure on a coordinate plane. There is not way to rotate the figure and have it map onto itself.

Thus, the figure has only line symmetry.

ANSWER: line

A figure has line symmetry if the figure can be Thus, the figure has both line symmetry and mapped onto itself by a reflection in a line. The rotational symmetry. given figure has 4 lines of symmetry. The line of symmetry are though the following pairs of points ANSWER: {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, line and rotational and {(2, 2), (2, –2)}. 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation Draw the figure on a coordinate plane. between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational

34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – A figure has line symmetry if the figure can be 2) mapped onto itself by a reflection in a line. The trapezoid has a line of reflection through points (0,3) SOLUTION: and (0, –3). Draw the figure on a coordinate plane. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. There is no way to rotate this figure and have it map onto itself. Thus, it does not have rotational symmetry.

Therefore, the figure has only line symmetry.

A figure has line symmetry if the figure can be ANSWER: mapped onto itself by a reflection in a line. The line given hexagon has 2 lines of symmetry. The lines pass through the following pair of points {(0, 4), (0, – ALGEBRA Graph the function and determine 4)}, and {(3, 0), (–3, 0)} whether the graph has line and/or rotational symmetry. If so, state the order and magnitude of A figure in the plane has rotational symmetry if the symmetry, and write the equations of any lines of figure can be mapped onto itself by a rotation symmetry. between 0° and 360° about the center of the figure. 36. y = x The figure has rotational symmetry. You can rotate the figure once within 360° and have it map to itself. SOLUTION: Graph the function. Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) SOLUTION: Draw the figure on a coordinate plane.

2 37. y = x + 1 SOLUTION: Graph the function.

A figure in the plane has rotational symmetry if the

figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. A figure has reflectional symmetry if the figure can The line can be rotated twice within 360° and be be mapped onto itself by a reflection in a line. The mapped onto itself. graph is reflected through the y-axis. Thus, the The number of times a figure maps onto itself as it equation of the line symmetry is x = 0. rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational A figure in the plane has rotational symmetry if the symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the The magnitude of symmetry is the smallest angle figure. There is no way to rotate the graph and have through which a figure can be rotated so that it maps it map onto itself. onto itself. The graph has magnitude of symmetry of Thus, the graph has only reflectional symmetry. . ANSWER:

line; x = 0 Thus, the graph has both reflectional and rotational symmetry.

ANSWER: rotational; 2; 180°; line symmetry; y = –x

38. y = –x3 SOLUTION: Graph the function.

2 37. y = x + 1 SOLUTION: Graph the function.

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself.

Thus, the graph has only rotational symmetry.

ANSWER: rotational; 2; 180°

38. y = –x3 SOLUTION: Graph the function.

39. Refer to the rectangle on the coordinate plane.

counterclockwise around its center of symmetry? A figure in the plane has rotational symmetry if the Explain. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the SOLUTION: figure. You can rotate the graph through the origin a. The lines of symmetry are parallel to the sides of and have it map onto itself. the rectangles, and through the center of rotation.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The slopes of the sides of the rectangle are 0.5 and symmetry. The graph has order 2 rotational -2, so the slopes of the lines of symmetry are the symmetry. same. The center of the rectangle is (1, 1.5). Use the The magnitude of symmetry is the smallest angle point-slope formula to find equations. through which a figure can be rotated so that it maps

onto itself. The graph has magnitude of symmetry of . b. The equations of the lines of symmetry do not Thus, the graph has only rotational symmetry. change; although the rectangle does not map onto ANSWER: itself under this rotation, the lines of symmetry are mapped to each other. rotational; 2; 180° ANSWER: a. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this

problem, you will use dynamic geometric software to 39. Refer to the rectangle on the coordinate plane. investigate line and rotational symmetry in regular polygons.

b. Geometric Use the rotation tool under the transformation menu to investigate the rotational a. What are the equations of the lines of symmetry of a the rectangle? symmetry of the figure in part . Then record its b. What happens to the equations of the lines of order of symmetry. c. Tabular Repeat the process in parts a and b for a symmetry when the rectangle is rotated 90 degrees square, regular pentagon, and regular hexagon. counterclockwise around its center of symmetry? Record the number of lines of symmetry and the Explain. order of symmetry for each polygon. SOLUTION: d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a a. The lines of symmetry are parallel to the sides of regular polygon with n sides. the rectangles, and through the center of rotation. SOLUTION: The slopes of the sides of the rectangle are 0.5 and a. Construct an equilateral triangle and label the -2, so the slopes of the lines of symmetry are the vertices A, B, and C. Draw a line through A same. perpendicular to . Reflect the triangle in the line. The center of the rectangle is (1, 1.5). Use the Show the labels of the reflected image. If the image point-slope formula to find equations. maps to the original, then this line is a line of reflection.

b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

ANSWER: a.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record Lastly, draw a line through C perpendicular to . their number. Reflect the triangle in the line. Show the labels of the b. Geometric Use the rotation tool under the reflected image. If the image maps to the original, transformation menu to investigate the rotational then this line is a line of reflection. symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides. SOLUTION: a. Construct an equilateral triangle and label the vertices A, B, and C. Draw a line through A perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

There are 3 lines of symmetry.

Next, draw a line through B perpendicular to . labels of the image. Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

There are 3 lines of symmetry.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Since the figure maps onto itself 3 times as it is rotated, the order of symmetry is 3. c. Square Construct a square and then construct lines through the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines Regular Hexagon constructed. So there are 4 lines of symmetry. Construct a regular hexagon and then construct lines Next, rotate the square about the center point. The through each vertex perpendicular to the sides. Use image maps to the original at 90, 180, 270, and 360 the reflection tool first to find that the image maps degree rotations. So the order of symmetry is 4. onto the original when reflected in each of the 6 lines constructed. So there are 6 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 60, 120, 180, 240, 300, and 360 degree rotations. So the order of symmetry is 6.

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5. d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and order of symmetry n.

ANSWER: a. 3 b. 3 c.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

A figure in the plane has rotational symmetry if the

figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines of symmetry and the order of symmetry. A regular

polygon with n sides has n lines of symmetry and The figure also has rotational symmetry. order of symmetry n. ANSWER: Therefore, neither of them are correct. Figure A has

a. 3 both line and rotational symmetry. b. 3 ANSWER: c. Neither; Figure A has both line and rotational symmetry.

42. CHALLENGE A quadrilateral in the coordinate

d. Sample answer: A regular polygon with n sides plane has exactly two lines of symmetry, y = x – 1 has n lines of symmetry and order of symmetry n. and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of 41. ERROR ANALYSIS Jaime says that Figure A has symmetry. only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them SOLUTION: correct? Explain your reasoning. Graph the figure and the lines of symmetry.

and b = . A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation A set of possible vertices for the figure are, (–1, 0), between 0° and 360° about the center of the figure. (2, 3), (4, 1), and (1, 2).

ANSWER: Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2)

The figure also has rotational symmetry.

Therefore, neither of them are correct. Figure A has both line and rotational symmetry.

ANSWER:

Neither; Figure A has both line and rotational symmetry. 43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. 42. CHALLENGE A quadrilateral in the coordinate plane has exactly two lines of symmetry, y = x – 1 SOLUTION: and y = –x + 2. Find a set of possible vertices for circle; Every line through the center of a circle is a the figure. Graph the figure and the lines of line of symmetry, and there are infinitely many such symmetry. lines. SOLUTION: ANSWER: Graph the figure and the lines of symmetry. circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can Pick points that are the same distance a from one be mapped onto itself by a rotation between 0° and line and the same distance b from the other line. In 360° about the center of the figure. the same answer, the quadrilateral is a rectangle with sides which are parallel to the lines of symmetry. Identify a figure that has line symmetry but does not This guarantees that the vertices of the quadrilateral have rotational symmetry. are the same distance a from one line and the same distance b from the other line. In this case, a = An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does and b = . not have rotational symmetry because it cannot be

rotated from 0° to 360° and map onto itself. A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2).

43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. SOLUTION: 45. WRITING IN MATH How are line symmetry and circle; Every line through the center of a circle is a rotational symmetry related? line of symmetry, and there are infinitely many such SOLUTION: lines. In both types of symmetries the figure is mapped ANSWER: onto itself. circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such Rotational symmetry. lines.

44. OPEN-ENDED Draw a figure with line symmetry Reflectional symmetry: but not rotational symmetry. Explain. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. In some cases an object can have both rotational and reflectional symmetry, such as the diamond, however Identify a figure that has line symmetry but does not some objects do not have both such as the crab. have rotational symmetry.

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

ANSWER: Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line ANSWER: symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is Sample answer: An isosceles triangle has line mapped onto itself by a rotation. A figure can have symmetry from the vertex angle to the base of the line symmetry and rotational symmetry. triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and 46. Sasha owns a tile store. For each tile in her store, she map onto itself. calculates the sum of the number of lines of symmetry and the order of symmetry, and then she enters this value into a database. Which value should she enter in the database for the tile shown here?

45. WRITING IN MATH How are line symmetry and rotational symmetry related? SOLUTION: In both types of symmetries the figure is mapped onto itself. A 2 B 3 Rotational symmetry. C 4 D 8

SOLUTION: Reflectional symmetry:

2 + 2 = 4, so C is the correct answer.

ANSWER: C

SOLUTION: A 2 Option A has rotational and reflectional symmetry. B 3 C 4 D 8 SOLUTION:

Option B has reflectional symmetry but not rotational symmetry.

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile.

Option C has neither rotational nor reflectional It has an order of symmetry of 2, because it has symmetry. rotational symmetry at 180 degrees, or each half turn around its center.

2 + 2 = 4, so C is the correct answer. Option D has rotational symmetry but not reflectional ANSWER: symmetry. C

The correct choice is D. D ANSWER: D

E 48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle SOLUTION: C Isosceles triangle Option A has rotational and reflectional symmetry. D Scalene triangle SOLUTION: An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

Option B has reflectional symmetry but not rotational symmetry. ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS

Option C has neither rotational nor reflectional has line symmetry but not rotational symmetry? symmetry. A B C D Option D has rotational symmetry but not reflectional symmetry. SOLUTION: First, plot the points.

Option E has reflectional symmetry but not rotational symmetry.

Then, plot each option A-D to consider each figure The correct choice is D. and its symmetry. ANSWER: Option A has both reflectional and rotational symmetry. D

ANSWER: C Option B has reflective symmetry but not rotational symmetry. The correct choice is B. 49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D SOLUTION: First, plot the points.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point. State whether the figure appears to have line symmetry. Write yes or no. If so, copy the Two lines of reflection go through the sides of the figure, draw all lines of symmetry, and state figure. their number.

SOLUTION: A figure has reflectional symmetry if the figure can Two lines of reflection go through the vertices of the be mapped onto itself by a reflection in a line. figure.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the

figure. Thus, there are four possible lines that go through the center and are lines of reflections.

Two lines of reflection go through the vertices of the figure.

Therefore, the figure has four lines of symmetry.

ANSWER: yes; 4

Thus, there are four possible lines that go through the center and are lines of reflections.

2. SOLUTION: Therefore, the figure has four lines of symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. ANSWER:

yes; 4 The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER: no

2. SOLUTION: A figure has reflectional symmetry if the figure can SOLUTION: be mapped onto itself by a reflection in a line. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The given figure has reflectional symmetry.

The given figure does not have reflectional The figure has a vertical line of symmetry. symmetry. There is no way to fold or reflect it onto itself.

ANSWER: no

It does not have a horizontal line of symmetry. 3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The figure does not have a line of symmetry through The given figure has reflectional symmetry. the vertices.

The figure has a vertical line of symmetry.

Thus, the figure has only one line of symmetry.

It does not have a horizontal line of symmetry. ANSWER: yes; 1

The figure does not have a line of symmetry through the vertices. State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

Thus, the figure has only one line of symmetry. 4. ANSWER: yes; 1 SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational State whether the figure has rotational symmetry. symmetry. Write yes or no. If so, copy the ANSWER: figure, locate the center of symmetry, and state no the order and magnitude of symmetry.

4. 5. SOLUTION: A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation A figure in the plane has rotational symmetry if the between 0° and 360° about the center of the figure. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure The given figure has rotational symmetry. were a regular pentagon, it would have rotational symmetry.

ANSWER: no

The number of times a figure maps onto itself as it 5. rotates form 0° and 360° is called the order of SOLUTION: symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The given figure has order of symmetry of 2, since between 0° and 360° about the center of the figure. the figure can be rotated twice in 360°.

The given figure has rotational symmetry. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is .

ANSWER: yes; 2; 180°

The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of symmetry.

The given figure has order of symmetry of 2, since the figure can be rotated twice in 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps 6. onto itself. SOLUTION: Since the figure has order 2 rotational symmetry, the A figure in the plane has rotational symmetry if the magnitude of the symmetry is . figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. ANSWER: yes; 2; 180° The given figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. 6. SOLUTION: Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The given figure has rotational symmetry.

The figure has magnitude of symmetry of .

ANSWER: yes; 4; 90°

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry State whether the figure has line symmetry and/or rotational symmetry. If so, describe the The magnitude of symmetry is the smallest angle reflections and/or rotations that map the figure through which a figure can be rotated so that it maps onto itself. onto itself.

The figure has magnitude of symmetry of .

ANSWER: yes; 4; 90° 7. SOLUTION: Vertical and horizontal lines through the center and diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way.

The equations of those lines in this figure are x = 0, State whether the figure has line symmetry y = -1, y = x - 1, and y = -x - 1. and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure Each quarter turn also maps the square onto itself. onto itself. So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the 7. rotations of 90, 180, and 270 degrees around the point SOLUTION: (0, -1) map the square onto itself. Vertical and horizontal lines through the center and diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way.

The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1.

Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around 8. the point (0, -1) map the square onto itself. SOLUTION: ANSWER: This figure does not have line symmetry, because adjacent sides are not congruent. line symmetry; rotational symmetry; the reflection in

the line x = 0, the reflection in the line y = -1, the It does have rotational symmetry for each half turn reflection in the line y = x - 1, and the reflection in around its center, so a rotation of 180 degrees around the line y = -x - 1 map the square onto itself; the the point (1, 1) maps the parallelogram onto itself. rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself. ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

REGULARITY State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number. 8. SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent. 9.

It does have rotational symmetry for each half turn SOLUTION: around its center, so a rotation of 180 degrees around A figure has reflectional symmetry if the figure can the point (1, 1) maps the parallelogram onto itself. be mapped onto itself by a reflection in a line.

ANSWER: For the given figure, there are no lines of reflection rotational symmetry; the rotation of 180 degrees where the figure can map onto itself. Thus, the figure around the point (1, 1) maps the parallelogram onto does not have any lines of of symmetry. itself. ANSWER: REGULARITY State whether the figure no appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

10. 9. SOLUTION: SOLUTION: A figure has reflectional line symmetry if the figure A figure has reflectional symmetry if the figure can can be mapped onto itself by a reflection in a line. be mapped onto itself by a reflection in a line. The given figure has reflectional symmetry. For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure In order for the figure to map onto itself, the line of does not have any lines of of symmetry. reflection must go through the center point.

ANSWER: The figure has a vertical and horizontal line of no reflection.

10.

SOLUTION: It is also possible to have reflection over the A figure has reflectional line symmetry if the figure diagonal lines. can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

The figure has a vertical and horizontal line of Therefore, the figure has four lines of symmetry reflection.

It is also possible to have reflection over the ANSWER: diagonal lines. yes; 4

Therefore, the figure has four lines of symmetry

11. SOLUTION: ANSWER: A figure has reflectional symmetry if the figure can yes; 4 be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though opposites edges.

11.

SOLUTION: There are three lines of reflection that go though A figure has reflectional symmetry if the figure can opposites vertices. be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though There are six possible lines that go through the center opposites edges. and are lines of reflections. Thus, the hexagon has six lines of symmetry.

There are three lines of reflection that go though opposites vertices.

ANSWER: yes; 6

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry.

12. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. ANSWER: yes; 6 The figure has reflectional symmetry.

There is only one line of symmetry, a horizontal line through the middle of the figure.

Thus, the figure has one line of symmetry.

ANSWER: 12. yes; 1 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

There is only one line of symmetry, a horizontal line 13. through the middle of the figure. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. Thus, the figure has one line of symmetry. There is only one possible line of reflection, ANSWER: horizontally though the middle of the figure. yes; 1

Thus, the figure has one line of symmetry.

ANSWER: yes; 1 13.

SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. 14. There is only one possible line of reflection, SOLUTION: horizontally though the middle of the figure. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional Thus, the figure has one line of symmetry. symmetry. It is not possible to draw a line of reflection where the figure can map onto itself. ANSWER: ANSWER: yes; 1 no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their number. 15. Refer to page 262. 14. SOLUTION: SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The flag does not have any reflectional symmetry. If

the red lines in the diagonals were in the same The given figure does not have reflectional location above and below the center horizontal line, symmetry. It is not possible to draw a line of the flag would have three lines of symmetry. reflection where the figure can map onto itself. ANSWER: ANSWER: no no

FLAGS State whether each flag design appears to 16. Refer to the flag on page 262. have line symmetry. Write yes or no. If so, copy the SOLUTION: flag, draw all lines of symmetry, and state their number. A figure has reflectional symmetry if the figure can 15. Refer to page 262. be mapped onto itself by a reflection in a line.

SOLUTION: The figure has reflectional symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. In order for the figure to map onto itself, the line of reflection must go through the center point. The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same A horizontal and vertical lines of reflection are location above and below the center horizontal line, possible. the flag would have three lines of symmetry.

ANSWER: no

16. Refer to the flag on page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

Two diagonal lines of reflection are possible. The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

There are a total of four possible lines that go through the center and are lines of reflections. Thus, the flag has four lines of symmetry.

Two diagonal lines of reflection are possible.

ANSWER: yes; 4

There are a total of four possible lines that go through the center and are lines of reflections. Thus, the flag has four lines of symmetry.

17. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line. ANSWER: yes; 4 The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

It is not possible to reflect over a vertical or line through the diagonals.

17. Refer to page 262. Thus, the figure has one line of symmetry. SOLUTION: ANSWER: A figure has reflectional symmetry if the figure can yes; 1 be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry. 18. SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the yes; 1 figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the The figure has rotational symmetry. figure, locate the center of symmetry, and state the order and magnitude of symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

18. This figure has order 2 rotational symmetry, since SOLUTION: you have to rotate 180° to get the figure to map onto itself. A figure in the plane has rotational symmetry if the

figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has a magnitude of symmetry of .

ANSWER: The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. yes; 2; 180° This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto itself.

The magnitude of symmetry is the smallest angle 19. through which a figure can be rotated so that it maps onto itself. SOLUTION: A figure in the plane has rotational symmetry if the The figure has a magnitude of symmetry of figure can be mapped onto itself by a rotation . between 0° and 360° about the center of the figure.

ANSWER: The triangle has rotational symmetry.

yes; 2; 180°

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of 19. symmetry. SOLUTION:

A figure in the plane has rotational symmetry if the The figure has order 3 rotational symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps

The triangle has rotational symmetry. onto itself.

The figure has magnitude of symmetry of .

ANSWER:

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of yes; 3; 120° symmetry.

The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle 20. through which a figure can be rotated so that it maps onto itself. SOLUTION: A figure in the plane has rotational symmetry if the The figure has magnitude of symmetry of figure can be mapped onto itself by a rotation . between 0° and 360° about the center of the figure.

ANSWER: The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

3-5 Symmetry ANSWER: yes; 3; 120° no

20. 21. SOLUTION: SOLUTION: A figure in the plane has rotational symmetry if the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. The crescent shaped figure has no rotational There is no way to rotate it such that it can be symmetry. There is no way to rotate it such that it mapped onto itself. can be mapped onto itself.

ANSWER: no

21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

ANSWER: no

22. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. eSolutions Manual - Powered by Cognero The figure has rotational symmetry. Page 9

ANSWER: no

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of 22. symmetry. SOLUTION: A figure in the plane has rotational symmetry if the The figure has order 8 rotational symmetry. This figure can be mapped onto itself by a rotation implies you can rotate the figure 8 times and have it between 0° and 360° about the center of the figure. map onto itself within 360°.

The figure has rotational symmetry. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This yes; 8; 45° implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps 23. onto itself. SOLUTION: The figure has magnitude of symmetry of A figure in the plane has rotational symmetry if the . figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. ANSWER: The figure has rotational symmetry.

yes; 8; 45°

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

23. The figure has order 8 rotational symmetry. This SOLUTION: means that the figure can be rotated 8 times and map A figure in the plane has rotational symmetry if the onto itself within 360°. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The figure has rotational symmetry. onto itself.

The figure has magnitude of symmetry of .

ANSWER:

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. yes; 8; 45° The figure has order 8 rotational symmetry. This WHEELS State whether each wheel cover appears means that the figure can be rotated 8 times and map to have rotational symmetry. Write yes or no. If so, onto itself within 360°. state the order and magnitude of symmetry.

24. Refer to page 263. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps SOLUTION: onto itself. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The figure has magnitude of symmetry of between 0° and 360° about the center of the figure. . The wheel has rotational symmetry. ANSWER: The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

yes; 8; 45° The wheel has order 5 rotational symmetry. There are 5 large spokes and 5 small spokes. You can WHEELS State whether each wheel cover appears rotate the wheel 5 times within 360° and map the to have rotational symmetry. Write yes or no. If so, figure onto itself. state the order and magnitude of symmetry. 24. Refer to page 263. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps SOLUTION: onto itself. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The wheel has magnitude of symmetry between 0° and 360° about the center of the figure. .

The wheel has rotational symmetry. ANSWER: yes; 5; 72° The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of 25. Refer to page 263. symmetry. SOLUTION: The wheel has order 5 rotational symmetry. There A figure in the plane has rotational symmetry if the are 5 large spokes and 5 small spokes. You can figure can be mapped onto itself by a rotation rotate the wheel 5 times within 360° and map the between 0° and 360° about the center of the figure. figure onto itself. The wheel has rotational symmetry. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The number of times a figure maps onto itself as it onto itself. rotates from 0° to 360° is called the order of symmetry. The wheel has magnitude of symmetry . The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times ANSWER: within 360° and map onto itself. yes; 5; 72° The magnitude of symmetry is the smallest angle 25. Refer to page 263. through which a figure can be rotated so that it maps onto itself. SOLUTION: A figure in the plane has rotational symmetry if the The wheel has order 8 rotational symmetry and figure can be mapped onto itself by a rotation magnitude . between 0° and 360° about the center of the figure. ANSWER: The wheel has rotational symmetry. yes; 8; 45°

The number of times a figure maps onto itself as it 26. Refer to page 263. rotates from 0° to 360° is called the order of SOLUTION: symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The wheel has order 8 rotational symmetry. There between 0° and 360° about the center of the figure. are 8 spokes, thus the wheel can be rotated 8 times The wheel has rotational symmetry. within 360° and map onto itself.

rotated 10 times within 360° and map onto itself. The wheel has order 8 rotational symmetry and . magnitude The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps onto itself. The wheel has magnitude of symmetry of yes; 8; 45° . 26. Refer to page 263. ANSWER: SOLUTION: yes; 10; 36° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation State whether the figure has line symmetry between 0° and 360° about the center of the figure. and/or rotational symmetry. If so, describe the The wheel has rotational symmetry. reflections and/or rotations that map the figure onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The wheel has order 10 rotational symmetry. There are 10 bolts and the tire can be rotated 10 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps 27. onto itself. The wheel has magnitude of symmetry of SOLUTION: . This triangle is scalene, so it cannot have symmetry.

ANSWER: ANSWER: yes; 10; 36° no symmetry State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

28. SOLUTION: This figure is a square, because each pair of adjacent sides is congruent and perpendicular. 27. All squares have both line and rotational symmetry. SOLUTION: The line symmetry is vertically, horizontally, and This triangle is scalene, so it cannot have symmetry. diagonally through the center of the square, with lines that are either parallel to the sides of the square or ANSWER: that include two vertices of the square. The no symmetry equations of those lines are: x = 0, y = 0, y = x, and y = -x

The rotational symmetry is for each quarter turn in a square, so the rotations of 90, 180, and 270 degrees around the origin map the square onto itself.

ANSWER: 28. line symmetry; rotational symmetry; the reflection in SOLUTION: the line x = 0, the reflection in the line y = 0, the This figure is a square, because each pair of adjacent reflection in the line y = x, and the reflection in the sides is congruent and perpendicular. line y = -x all map the square onto itself; the rotations All squares have both line and rotational symmetry. of 90, 180, and 270 degrees around the origin map

The line symmetry is vertically, horizontally, and the square onto itself. diagonally through the center of the square, with lines that are either parallel to the sides of the square or that include two vertices of the square. The equations of those lines are: x = 0, y = 0, y = x, and y = -x

The rotational symmetry is for each quarter turn in a square, so the rotations of 90, 180, and 270 degrees 29. around the origin map the square onto itself. SOLUTION: The trapezoid has line symmetry, because it is ANSWER: isosceles, but it does not have rotational symmetry, because no trapezoid does. line symmetry; rotational symmetry; the reflection in

the line x = 0, the reflection in the line y = 0, the The reflection in the line y = 1.5 maps the trapezoid reflection in the line y = x, and the reflection in the onto itself, because that is the perpendicular bisector line y = -x all map the square onto itself; the rotations to the parallel sides. of 90, 180, and 270 degrees around the origin map the square onto itself. ANSWER: line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself.

29. SOLUTION: The trapezoid has line symmetry, because it is 30. isosceles, but it does not have rotational symmetry, because no trapezoid does. SOLUTION: This figure is a parallelogram, so it has rotational The reflection in the line y = 1.5 maps the trapezoid symmetry of a half turn or 180 degrees around its onto itself, because that is the perpendicular bisector center, which is the point (1, -1.5). to the parallel sides. Since this parallelogram is not a rhombus it does not ANSWER: have line symmetry. line symmetry; the reflection in the line y = 1.5 maps ANSWER: the trapezoid onto itself. rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram onto itself.

31. MODELING Symmetry is an important component of photography. Photographers often use reflection in water to create symmetry in photos. The photo on page 263 is a long exposure shot of the Eiffel tower 30. reflected in a pool. SOLUTION: This figure is a parallelogram, so it has rotational a. Describe the two-dimensional symmetry created symmetry of a half turn or 180 degrees around its by the photo. center, which is the point (1, -1.5). b. Is there rotational symmetry in the photo? Explain your reasoning. Since this parallelogram is not a rhombus it does not SOLUTION: have line symmetry. a Sample answer: There is a horizontal line of ANSWER: symmetry between the tower and its reflection. rotational symmetry; the rotation of 180 degrees There is a vertical line of symmetry through the around the point (1, -1.5) maps the parallelogram center of the photo. onto itself. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no 31. MODELING Symmetry is an important component rotational symmetry. of photography. Photographers often use reflection in water to create symmetry in photos. The photo on ANSWER: page 263 is a long exposure shot of the Eiffel tower a. Sample answer: There is a horizontal line of reflected in a pool. symmetry between the tower and its reflection. There is a vertical line of symmetry through the a. Describe the two-dimensional symmetry created center of the photo. by the photo. b No; sample answer: Because of how the image is b. Is there rotational symmetry in the photo? Explain reflected over the horizontal line, there is no your reasoning. rotational symmetry. SOLUTION: COORDINATE GEOMETRY Determine a Sample answer: There is a horizontal line of whether the figure with the given vertices has line symmetry between the tower and its reflection. symmetry and/or rotational symmetry. There is a vertical line of symmetry through the 32. R(–3, 3), S(–3, –3), T(3, 3)

center of the photo. SOLUTION: b No; sample answer: Because of how the image is Draw the figure on a coordinate plane. reflected over the horizontal line, there is no rotational symmetry.

ANSWER: a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no rotational symmetry. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. COORDINATE GEOMETRY Determine whether the figure with the given vertices has line The given triangle has a line of symmetry through symmetry and/or rotational symmetry. points (0, 0) and (–3, 3). 32. R(–3, 3), S(–3, –3), T(3, 3) SOLUTION: A figure in the plane has rotational symmetry if the Draw the figure on a coordinate plane. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. There is not way to rotate the figure and have it map onto itself.

Thus, the figure has only line symmetry.

ANSWER: line

A figure has line symmetry if the figure can be 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) mapped onto itself by a reflection in a line. SOLUTION:

Draw the figure on a coordinate plane. The given triangle has a line of symmetry through points (0, 0) and (–3, 3).

Thus, the figure has only line symmetry. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The ANSWER: given figure has 4 lines of symmetry. The line of line symmetry are though the following pairs of points {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) and {(2, 2), (2, –2)}.

SOLUTION: A figure in the plane has rotational symmetry if the Draw the figure on a coordinate plane. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – given figure has 4 lines of symmetry. The line of 2) symmetry are though the following pairs of points SOLUTION: {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, and {(2, 2), (2, –2)}. Draw the figure on a coordinate plane.

Thus, the figure has both line symmetry and rotational symmetry. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The ANSWER: given hexagon has 2 lines of symmetry. The lines line and rotational pass through the following pair of points {(0, 4), (0, – 4)}, and {(3, 0), (–3, 0)} 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, –

2) A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation Draw the figure on a coordinate plane. between 0° and 360° about the center of the figure. The figure has rotational symmetry. You can rotate the figure once within 360° and have it map to itself.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational

A figure has line symmetry if the figure can be 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) mapped onto itself by a reflection in a line. The SOLUTION: given hexagon has 2 lines of symmetry. The lines pass through the following pair of points {(0, 4), (0, – Draw the figure on a coordinate plane. 4)}, and {(3, 0), (–3, 0)}

35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation between 0° and 360° about the center of the Draw the figure on a coordinate plane. figure. There is no way to rotate this figure and have it map onto itself. Thus, it does not have rotational symmetry.

Therefore, the figure has only line symmetry.

ANSWER: line

ALGEBRA Graph the function and determine A figure has line symmetry if the figure can be whether the graph has line and/or rotational mapped onto itself by a reflection in a line. The symmetry. If so, state the order and magnitude of trapezoid has a line of reflection through points (0,3) symmetry, and write the equations of any lines of and (0, –3). symmetry. 36. y = x A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation Graph the function. between 0° and 360° about the center of the figure. There is no way to rotate this figure and have it map onto itself. Thus, it does not have rotational symmetry.

Therefore, the figure has only line symmetry.

ANSWER: line

ALGEBRA Graph the function and determine whether the graph has line and/or rotational A figure has reflectional symmetry if the figure can symmetry. If so, state the order and magnitude of be mapped onto itself by a reflection in a line. The symmetry, and write the equations of any lines of line y = x has reflectional symmetry since any line symmetry. perpendicular to y = x is a line of reflection. The 36. y = x equation of the line symmetry is y = –x. SOLUTION: Graph the function. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The line can be rotated twice within 360° and be mapped onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry.

The magnitude of symmetry is the smallest angle A figure has reflectional symmetry if the figure can through which a figure can be rotated so that it maps be mapped onto itself by a reflection in a line. The onto itself. line y = x has reflectional symmetry since any line The graph has magnitude of symmetry of perpendicular to y = x is a line of reflection. The . equation of the line symmetry is y = –x. Thus, the graph has both reflectional and rotational A figure in the plane has rotational symmetry if the symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. ANSWER: The line can be rotated twice within 360° and be rotational; 2; 180°; line symmetry; y = –x mapped onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of 2 . 37. y = x + 1

SOLUTION: Thus, the graph has both reflectional and rotational symmetry. Graph the function.

ANSWER: rotational; 2; 180°; line symmetry; y = –x

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The

graph is reflected through the y-axis. Thus, the equation of the line symmetry is x = 0. 2 37. y = x + 1 A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation Graph the function. between 0° and 360° about the center of the figure. There is no way to rotate the graph and have it map onto itself.

Thus, the graph has only reflectional symmetry.

ANSWER: line; x = 0

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph is reflected through the y-axis. Thus, the equation of the line symmetry is x = 0.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the 38. y = –x3 figure. There is no way to rotate the graph and have it map onto itself. SOLUTION: Graph the function. Thus, the graph has only reflectional symmetry.

ANSWER: line; x = 0

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself. 38. y = –x3 A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation Graph the function. between 0° and 360° about the center of the figure. You can rotate the graph through the origin and have it map onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps A figure has reflectional symmetry if the figure can onto itself. The graph has magnitude of symmetry of be mapped onto itself by a reflection in a line. The . graph does not have a line of reflections where the

graph can be mapped onto itself. Thus, the graph has only rotational symmetry. A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation rotational; 2; 180° between 0° and 360° about the center of the figure. You can rotate the graph through the origin and have it map onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps 39. Refer to the rectangle on the coordinate plane. onto itself. The graph has magnitude of symmetry of .

Thus, the graph has only rotational symmetry.

ANSWER: rotational; 2; 180°

a. What are the equations of the lines of symmetry of the rectangle? b. What happens to the equations of the lines of symmetry when the rectangle is rotated 90 degrees counterclockwise around its center of symmetry? Explain. SOLUTION: a. The lines of symmetry are parallel to the sides of 39. Refer to the rectangle on the coordinate plane. the rectangles, and through the center of rotation.

The slopes of the sides of the rectangle are 0.5 and -2, so the slopes of the lines of symmetry are the same. The center of the rectangle is (1, 1.5). Use the point-slope formula to find equations.

a. What are the equations of the lines of symmetry of the rectangle?

b. What happens to the equations of the lines of b. The equations of the lines of symmetry do not symmetry when the rectangle is rotated 90 degrees change; although the rectangle does not map onto counterclockwise around its center of symmetry? itself under this rotation, the lines of symmetry are Explain. mapped to each other. SOLUTION: a. The lines of symmetry are parallel to the sides of ANSWER: the rectangles, and through the center of rotation. a. b. The equations of the lines of symmetry do not The slopes of the sides of the rectangle are 0.5 and change; although the rectangle does not map onto -2, so the slopes of the lines of symmetry are the itself under this rotation, the lines of symmetry are same. mapped to each other. The center of the rectangle is (1, 1.5). Use the 40. MULTIPLE REPRESENTATIONS In this point-slope formula to find equations. problem, you will use dynamic geometric software to

investigate line and rotational symmetry in regular polygons.

a. Geometric b. The equations of the lines of symmetry do not Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool change; although the rectangle does not map onto under the transformation menu to investigate and itself under this rotation, the lines of symmetry are determine all possible lines of symmetry. Then record mapped to each other. their number. b. Geometric Use the rotation tool under the ANSWER: transformation menu to investigate the rotational a. symmetry of the figure in part a. Then record its b. The equations of the lines of symmetry do not order of symmetry. change; although the rectangle does not map onto c. Tabular Repeat the process in parts a and b for a itself under this rotation, the lines of symmetry are square, regular pentagon, and regular hexagon. mapped to each other. Record the number of lines of symmetry and the order of symmetry for each polygon. 40. MULTIPLE REPRESENTATIONS In this d. Verbal Make a conjecture about the number of problem, you will use dynamic geometric software to lines of symmetry and the order of symmetry for a investigate line and rotational symmetry in regular regular polygon with n sides. polygons. SOLUTION:

a. Geometric Use The Geometer’s Sketchpad to a. Construct an equilateral triangle and label the draw an equilateral triangle. Use the reflection tool vertices A, B, and C. Draw a line through A under the transformation menu to investigate and perpendicular to . Reflect the triangle in the line. determine all possible lines of symmetry. Then record Show the labels of the reflected image. If the image their number. maps to the original, then this line is a line of b. Geometric Use the rotation tool under the reflection. transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides.

SOLUTION: Next, draw a line through B perpendicular to . a. Construct an equilateral triangle and label the Reflect the triangle in the line. Show the labels of the vertices A, B, and C. Draw a line through A reflected image. If the image maps to the original, perpendicular to . Reflect the triangle in the line. then this line is a line of reflection. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection. There are 3 lines of symmetry.

There are 3 lines of symmetry.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original. Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Since the figure maps onto itself 3 times as it is rotated, the order of symmetry is 3. The triangle can be rotated a third time about D. A c. 360 degree rotation maps the image to the original. Square Construct a square and then construct lines through the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Since the figure maps onto itself 3 times as it is rotated, the order of symmetry is 3. c. Square Construct a square and then construct lines through the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The Regular Pentagon image maps to the original at 90, 180, 270, and 360 Construct a regular pentagon and then construct lines degree rotations. So the order of symmetry is 4. through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5.

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and Regular Hexagon 360 degree rotations. So the order of symmetry is 5. Construct a regular hexagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 6 lines constructed. So there are 6 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 60, 120, 180, 240, 300, and 360 degree rotations. So the order of symmetry is 6.

Regular Hexagon Construct a regular hexagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 6 lines constructed. So there are 6 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 60, 120, 180, 240, 300, and 360 degree rotations. So the order of symmetry is 6. d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and order of symmetry n.

ANSWER: a. 3 b. 3 c.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A d. Sample answer: for each figure studied, the has only rotational symmetry. Is either of them number of sides of the figure is the same as the lines correct? Explain your reasoning. of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and order of symmetry n.

ANSWER: a. 3 b. 3 c. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. This figure has 4 lines of symmetry.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. This figure has 4 lines of symmetry. The figure also has rotational symmetry.

Therefore, neither of them are correct. Figure A has both line and rotational symmetry.

ANSWER: Neither; Figure A has both line and rotational symmetry.

A figure in the plane has rotational symmetry if the 42. CHALLENGE A quadrilateral in the coordinate figure can be mapped onto itself by a rotation plane has exactly two lines of symmetry, y = x – 1 between 0° and 360° about the center of the figure. and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of symmetry. SOLUTION: Graph the figure and the lines of symmetry.

The figure also has rotational symmetry.

Therefore, neither of them are correct. Figure A has both line and rotational symmetry.

ANSWER: Neither; Figure A has both line and rotational symmetry. Pick points that are the same distance a from one 42. CHALLENGE A quadrilateral in the coordinate line and the same distance b from the other line. In plane has exactly two lines of symmetry, y = x – 1 the same answer, the quadrilateral is a rectangle with and y = –x + 2. Find a set of possible vertices for sides which are parallel to the lines of symmetry. the figure. Graph the figure and the lines of This guarantees that the vertices of the quadrilateral symmetry. are the same distance a from one line and the same SOLUTION: distance b from the other line. In this case, a = Graph the figure and the lines of symmetry. and b = .

A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2).

ANSWER: Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2)

distance b from the other line. In this case, a = 43. REASONING A figure has infinitely many lines of and b = . symmetry. What is the figure? Explain.

SOLUTION: A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2). circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such ANSWER: lines. Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) ANSWER: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure 43. REASONING A figure has infinitely many lines of in the plane has rotational symmetry if the figure can symmetry. What is the figure? Explain. be mapped onto itself by a rotation between 0° and 360° about the center of the figure. SOLUTION:

circle; Every line through the center of a circle is a Identify a figure that has line symmetry but does not line of symmetry, and there are infinitely many such have rotational symmetry. lines. ANSWER: An isosceles triangle has line symmetry from the circle; Every line through the center of a circle is a vertex angle to the base of the triangle, but it does line of symmetry, and there are infinitely many such not have rotational symmetry because it cannot be

lines. rotated from 0° to 360° and map onto itself.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain.

SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure ANSWER: in the plane has rotational symmetry if the figure can Sample answer: An isosceles triangle has line be mapped onto itself by a rotation between 0° and symmetry from the vertex angle to the base of the 360° about the center of the figure. triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and Identify a figure that has line symmetry but does not map onto itself. have rotational symmetry.

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself. 45. WRITING IN MATH How are line symmetry and rotational symmetry related? SOLUTION: In both types of symmetries the figure is mapped onto itself. ANSWER: Rotational symmetry. Sample answer: An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and Reflectional symmetry: map onto itself.

45. WRITING IN MATH How are line symmetry and In some cases an object can have both rotational and rotational symmetry related? reflectional symmetry, such as the diamond, however some objects do not have both such as the crab. SOLUTION: In both types of symmetries the figure is mapped onto itself.

Rotational symmetry.

Reflectional symmetry: ANSWER: Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry. In some cases an object can have both rotational and reflectional symmetry, such as the diamond, however 46. Sasha owns a tile store. For each tile in her store, she some objects do not have both such as the crab. calculates the sum of the number of lines of symmetry and the order of symmetry, and then she enters this value into a database. Which value should she enter in the database for the tile shown here?

ANSWER: Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line A 2 symmetry the figure is mapped onto itself by a B 3 reflection, and in rotational symmetry, a figure is C 4 mapped onto itself by a rotation. A figure can have D 8 line symmetry and rotational symmetry. SOLUTION: 46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of symmetry and the order of symmetry, and then she enters this value into a database. Which value should she enter in the database for the tile shown here?

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn around its center.

A 2 2 + 2 = 4, so C is the correct answer. B 3 ANSWER: C 4 D 8 C SOLUTION: 47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A

The tile is a rhombus and has 2 lines of symmetry. B Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has

rotational symmetry at 180 degrees, or each half turn C around its center.

2 + 2 = 4, so C is the correct answer. D ANSWER: C E 47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A SOLUTION: Option A has rotational and reflectional symmetry.

Option B has reflectional symmetry but not rotational C symmetry.

E Option C has neither rotational nor reflectional symmetry.

SOLUTION: Option A has rotational and reflectional symmetry. Option D has rotational symmetry but not reflectional symmetry.

Option B has reflectional symmetry but not rotational symmetry.

Option E has reflectional symmetry but not rotational symmetry.

Option C has neither rotational nor reflectional symmetry.

The correct choice is D.

Option D has rotational symmetry but not reflectional ANSWER: symmetry. D

48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle Option E has reflectional symmetry but not rotational C Isosceles triangle symmetry. D Scalene triangle SOLUTION: An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

ANSWER: C The correct choice is D. 49. Camryn plotted the points , ANSWER: and . Which of the following additional points D can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry? 48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A A Equilateral triangle B B Equiangular triangle C C Isosceles triangle D D Scalene triangle SOLUTION: SOLUTION: First, plot the points. An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C Then, plot each option A-D to consider each figure D and its symmetry. Option A has both reflectional and rotational SOLUTION: symmetry. First, plot the points.

Then, plot each option A-D to consider each figure Option B has reflective symmetry but not rotational and its symmetry. symmetry. The correct choice is B. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

Thus, there are four possible lines that go through the center and are lines of reflections.

State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

Therefore, the figure has four lines of symmetry.

ANSWER: 1. yes; 4 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure. 2. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto Two lines of reflection go through the vertices of the itself. figure. ANSWER: no

The given figure has reflectional symmetry.

The figure has a vertical line of symmetry.

Therefore, the figure has four lines of symmetry.

ANSWER: yes; 4 It does not have a horizontal line of symmetry.

The figure does not have a line of symmetry through the vertices.

2. SOLUTION:

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. Thus, the figure has only one line of symmetry.

The given figure does not have reflectional ANSWER: symmetry. There is no way to fold or reflect it onto yes; 1 itself.

ANSWER: no

The figure does not have a line of symmetry through the vertices.

5. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Thus, the figure has only one line of symmetry. The given figure has rotational symmetry.

ANSWER: yes; 1

The magnitude of symmetry is the smallest angle

For the given figure, there is no rotation between 0° ANSWER: and 360° that maps the figure onto itself. If the figure yes; 2; 180° were a regular pentagon, it would have rotational symmetry.

ANSWER: no

The given figure has rotational symmetry. The given figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The number of times a figure maps onto itself as it symmetry.

rotates form 0° and 360° is called the order of symmetry. Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

Since the figure has order 2 rotational symmetry, the ANSWER: magnitude of the symmetry is . yes; 4; 90° ANSWER: yes; 2; 180°

State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

6. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. 7.

The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1.

The figure has magnitude of symmetry of .

ANSWER: yes; 4; 90°

8. SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent.

10. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

9. SOLUTION: Therefore, the figure has four lines of symmetry A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry.

ANSWER: no ANSWER: yes; 4

10. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

In order for the figure to map onto itself, the line of

reflection must go through the center point. 11.

The figure has a vertical and horizontal line of SOLUTION: reflection. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point.

It is also possible to have reflection over the There are three lines of reflection that go though diagonal lines. opposites edges.

Therefore, the figure has four lines of symmetry There are three lines of reflection that go though opposites vertices.

ANSWER: There are six possible lines that go through the center yes; 4 and are lines of reflections. Thus, the hexagon has six lines of symmetry.

ANSWER: yes; 6 11. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

The figure has reflectional symmetry.

There is only one line of symmetry, a horizontal line through the middle of the figure. There are three lines of reflection that go though opposites vertices.

Thus, the figure has one line of symmetry.

ANSWER:

yes; 1 There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry.

13. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

ANSWER: The figure has reflectional symmetry. yes; 6 There is only one possible line of reflection, horizontally though the middle of the figure.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

12. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. 14.

There is only one line of symmetry, a horizontal line SOLUTION: through the middle of the figure. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

Thus, the figure has one line of symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. ANSWER: yes; 1 The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

14. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. It is not possible to draw a line of reflection where the figure can map onto itself.

ANSWER: Two diagonal lines of reflection are possible. no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their number. 15. Refer to page 262. SOLUTION:

A figure has reflectional symmetry if the figure can

ANSWER: no

16. Refer to the flag on page 262. SOLUTION: A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line. ANSWER:

yes; 4 The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

17. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

Two diagonal lines of reflection are possible. The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

It is not possible to reflect over a vertical or line

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state ANSWER: the order and magnitude of symmetry. yes; 4

18. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

17. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The figure has rotational symmetry.

This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto itself.

The magnitude of symmetry is the smallest angle It is not possible to reflect over a vertical or line through which a figure can be rotated so that it maps through the diagonals. onto itself.

Thus, the figure has one line of symmetry. The figure has a magnitude of symmetry of . ANSWER: ANSWER: yes; 1

yes; 2; 180°

A figure in the plane has rotational symmetry if the The triangle has rotational symmetry. figure can be mapped onto itself by a rotation

between 0° and 360° about the center of the figure.

symmetry.

The figure has order 3 rotational symmetry.

onto itself.

ANSWER:

yes; 3; 120°

yes; 2; 180°

The triangle has rotational symmetry. ANSWER: no

21.

The figure has magnitude of symmetry of .

ANSWER:

yes; 3; 120°

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself. ANSWER: no ANSWER: no

The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

3-5 SymmetryANSWER: no yes; 8; 45°

The figure has rotational symmetry. The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of

ANSWER:

yes; 8; 45°

A figure in the plane has rotational symmetry if the The wheel has rotational symmetry. eSolutionsfigureManual can be- Powered mappedby Cogneroonto itself by a rotation Page 10 between 0° and 360° about the center of the figure. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of

The figure has rotational symmetry. symmetry.

The wheel has order 5 rotational symmetry. There are 5 large spokes and 5 small spokes. You can rotate the wheel 5 times within 360° and map the figure onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of

yes; 8; 45° symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself. 27.

SOLUTION: The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps This triangle is scalene, so it cannot have symmetry.

onto itself. ANSWER:

no symmetry The wheel has order 8 rotational symmetry and magnitude .

ANSWER: yes; 8; 45° 26. Refer to page 263.

The reflection in the line y = 1.5 maps the trapezoid onto itself, because that is the perpendicular bisector to the parallel sides.

Since this parallelogram is not a rhombus it does not have line symmetry.

ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram onto itself.

COORDINATE GEOMETRY Determine

Thus, the figure has only line symmetry.

ANSWER: line

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational

ANSWER: line and rotational 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) SOLUTION: Draw the figure on a coordinate plane.

2 37. y = x + 1 SOLUTION: Graph the function.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The line can be rotated twice within 360° and be A figure has reflectional symmetry if the figure can mapped onto itself. be mapped onto itself by a reflection in a line. The graph is reflected through the y-axis. Thus, the The number of times a figure maps onto itself as it equation of the line symmetry is x = 0. rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational A figure in the plane has rotational symmetry if the symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the The magnitude of symmetry is the smallest angle figure. There is no way to rotate the graph and have through which a figure can be rotated so that it maps it map onto itself. onto itself. The graph has magnitude of symmetry of Thus, the graph has only reflectional symmetry. . ANSWER: Thus, the graph has both reflectional and rotational line; x = 0 symmetry.

ANSWER: rotational; 2; 180°; line symmetry; y = –x

38. y = –x3 SOLUTION: Graph the function.

2 37. y = x + 1 SOLUTION: Graph the function.

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself.

Thus, the graph has only rotational symmetry.

ANSWER: rotational; 2; 180°

38. y = –x3 SOLUTION: Graph the function.

39. Refer to the rectangle on the coordinate plane.

b. The equations of the lines of symmetry do not Thus, the graph has only rotational symmetry. change; although the rectangle does not map onto ANSWER: itself under this rotation, the lines of symmetry are rotational; 2; 180° mapped to each other. ANSWER: a. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. Lastly, draw a line through C perpendicular to . b. Geometric Use the rotation tool under the Reflect the triangle in the line. Show the labels of the transformation menu to investigate the rotational reflected image. If the image maps to the original, symmetry of the figure in part a. Then record its then this line is a line of reflection. order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides. SOLUTION: a. Construct an equilateral triangle and label the vertices A, B, and C. Draw a line through A perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

There are 3 lines of symmetry.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

There are 3 lines of symmetry.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Regular Pentagon

Since the figure maps onto itself 3 times as it is rotated, the order of symmetry is 3. c. Square Construct a square and then construct lines through the midpoints of each side and diagonals. Use the

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5. d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and order of symmetry n.

ANSWER: a. 3 b. 3 c.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

is 6. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. This figure has 4 lines of symmetry.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and

order of symmetry n. The figure also has rotational symmetry.

ANSWER: Therefore, neither of them are correct. Figure A has a. 3 both line and rotational symmetry. b. 3 ANSWER: c. Neither; Figure A has both line and rotational symmetry.

The figure also has rotational symmetry.

Therefore, neither of them are correct. Figure A has both line and rotational symmetry.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can Pick points that are the same distance a from one be mapped onto itself by a rotation between 0° and line and the same distance b from the other line. In 360° about the center of the figure. the same answer, the quadrilateral is a rectangle with sides which are parallel to the lines of symmetry. Identify a figure that has line symmetry but does not This guarantees that the vertices of the quadrilateral have rotational symmetry. are the same distance a from one line and the same distance b from the other line. In this case, a = An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does and b = . not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself. A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2).

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

ANSWER: Sample answer: In both rotational and line symmetry

45. WRITING IN MATH How are line symmetry and rotational symmetry related?

SOLUTION: In both types of symmetries the figure is mapped onto itself. A 2 B 3 Rotational symmetry. C 4 D 8 SOLUTION: Reflectional symmetry:

2 + 2 = 4, so C is the correct answer.

ANSWER: C

SOLUTION: A 2 B 3 Option A has rotational and reflectional symmetry. C 4 D 8 SOLUTION:

Option B has reflectional symmetry but not rotational symmetry.

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has Option C has neither rotational nor reflectional rotational symmetry at 180 degrees, or each half turn symmetry. around its center.

2 + 2 = 4, so C is the correct answer.

ANSWER: Option D has rotational symmetry but not reflectional C symmetry.

Option B has reflectional symmetry but not rotational symmetry. ANSWER: C

Option E has reflectional symmetry but not rotational symmetry.

The correct choice is D. Then, plot each option A-D to consider each figure and its symmetry. ANSWER: Option A has both reflectional and rotational D symmetry.

ANSWER:

A B C D SOLUTION: First, plot the points.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure. State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

Thus, there are four possible lines that go through 1. the center and are lines of reflections. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of

reflection must go through the center point.

Therefore, the figure has four lines of symmetry. Two lines of reflection go through the sides of the figure. ANSWER: yes; 4

Two lines of reflection go through the vertices of the figure.

2. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. Thus, there are four possible lines that go through the center and are lines of reflections. The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER: no

Therefore, the figure has four lines of symmetry. 3. ANSWER: SOLUTION: yes; 4 A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

The figure has a vertical line of symmetry.

2. SOLUTION: It does not have a horizontal line of symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional

symmetry. There is no way to fold or reflect it onto itself. The figure does not have a line of symmetry through the vertices. ANSWER: no

3. Thus, the figure has only one line of symmetry. SOLUTION: A figure has reflectional symmetry if the figure can ANSWER: be mapped onto itself by a reflection in a line. yes; 1

The given figure has reflectional symmetry.

The figure has a vertical line of symmetry.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state

the order and magnitude of symmetry. It does not have a horizontal line of symmetry.

SOLUTION: The figure does not have a line of symmetry through A figure in the plane has rotational symmetry if the the vertices. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: Thus, the figure has only one line of symmetry. no ANSWER: yes; 1

5. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation State whether the figure has rotational between 0° and 360° about the center of the figure. symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state The given figure has rotational symmetry. the order and magnitude of symmetry.

4. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of For the given figure, there is no rotation between 0° symmetry. and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational The given figure has order of symmetry of 2, since symmetry. the figure can be rotated twice in 360°. ANSWER: no The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is . 5. ANSWER: SOLUTION: yes; 2; 180° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

6. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The number of times a figure maps onto itself as it between 0° and 360° about the center of the figure. rotates form 0° and 360° is called the order of symmetry. The given figure has rotational symmetry.

The given figure has order of symmetry of 2, since the figure can be rotated twice in 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The number of times a figure maps onto itself as it Since the figure has order 2 rotational symmetry, the rotates from 0° to 360° is called the order of magnitude of the symmetry is . symmetry.

ANSWER: Since the figure can be rotated 4 times within 360° , yes; 2; 180° it has order 4 rotational symmetry

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER: yes; 4; 90° 6. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry. State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. 7. Since the figure can be rotated 4 times within 360° , SOLUTION: it has order 4 rotational symmetry Vertical and horizontal lines through the center and diagonal lines through two opposite vertices are all The magnitude of symmetry is the smallest angle lines of symmetry for a square oriented this way. through which a figure can be rotated so that it maps onto itself. The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1. The figure has magnitude of symmetry of . Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around ANSWER: the point (0, -1) map the square onto itself. yes; 4; 90° ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself. State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

8. SOLUTION: 7. This figure does not have line symmetry, because adjacent sides are not congruent. SOLUTION: Vertical and horizontal lines through the center and It does have rotational symmetry for each half turn diagonal lines through two opposite vertices are all around its center, so a rotation of 180 degrees around lines of symmetry for a square oriented this way. the point (1, 1) maps the parallelogram onto itself.

The equations of those lines in this figure are x = 0, ANSWER: y = -1, y = x - 1, and y = -x - 1. rotational symmetry; the rotation of 180 degrees around the point (1, 1) maps the parallelogram onto Each quarter turn also maps the square onto itself. itself. So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself. REGULARITY State whether the figure appears to have line symmetry. Write yes or no. ANSWER: If so, copy the figure, draw all lines of line symmetry; rotational symmetry; the reflection in symmetry, and state their number. the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point 9. (0, -1) map the square onto itself. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry.

8. ANSWER: no SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself. 10.

ANSWER: SOLUTION: rotational symmetry; the rotation of 180 degrees A figure has reflectional line symmetry if the figure around the point (1, 1) maps the parallelogram onto can be mapped onto itself by a reflection in a line. itself. The given figure has reflectional symmetry. REGULARITY State whether the figure appears to have line symmetry. Write yes or no. In order for the figure to map onto itself, the line of If so, copy the figure, draw all lines of reflection must go through the center point. symmetry, and state their number. The figure has a vertical and horizontal line of reflection.

9. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection It is also possible to have reflection over the where the figure can map onto itself. Thus, the figure diagonal lines. does not have any lines of of symmetry.

ANSWER: no

Therefore, the figure has four lines of symmetry

10. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry. ANSWER: In order for the figure to map onto itself, the line of yes; 4 reflection must go through the center point.

The figure has a vertical and horizontal line of reflection.

It is also possible to have reflection over the diagonal lines. 11. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry. Therefore, the figure has four lines of symmetry In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though opposites edges.

ANSWER: yes; 4

There are three lines of reflection that go though opposites vertices.

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has 11. six lines of symmetry. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point. ANSWER: There are three lines of reflection that go though yes; 6 opposites edges.

There are three lines of reflection that go though opposites vertices.

12. SOLUTION: A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line.

There are six possible lines that go through the center The figure has reflectional symmetry. and are lines of reflections. Thus, the hexagon has six lines of symmetry. There is only one line of symmetry, a horizontal line through the middle of the figure.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1 ANSWER: yes; 6

13.

SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

12. The figure has reflectional symmetry. SOLUTION: A figure has reflectional symmetry if the figure can There is only one possible line of reflection, be mapped onto itself by a reflection in a line. horizontally though the middle of the figure.

The figure has reflectional symmetry.

There is only one line of symmetry, a horizontal line Thus, the figure has one line of symmetry. through the middle of the figure. ANSWER: yes; 1

Thus, the figure has one line of symmetry.

ANSWER: yes; 1 14. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional 13. symmetry. It is not possible to draw a line of reflection where the figure can map onto itself. SOLUTION: A figure has reflectional symmetry if the figure can ANSWER: be mapped onto itself by a reflection in a line. no

The figure has reflectional symmetry. FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the There is only one possible line of reflection, flag, draw all lines of symmetry, and state their

horizontally though the middle of the figure. number. 15. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can Thus, the figure has one line of symmetry. be mapped onto itself by a reflection in a line.

ANSWER: The flag does not have any reflectional symmetry. If yes; 1 the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry. ANSWER: no

16. Refer to the flag on page 262. 14. SOLUTION: SOLUTION: A figure has reflectional symmetry if the figure can A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. be mapped onto itself by a reflection in a line. The figure has reflectional symmetry. The given figure does not have reflectional symmetry. It is not possible to draw a line of In order for the figure to map onto itself, the line of reflection where the figure can map onto itself. reflection must go through the center point.

ANSWER: A horizontal and vertical lines of reflection are no possible. FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their number. 15. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The flag does not have any reflectional symmetry. If Two diagonal lines of reflection are possible. the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER: no

16. Refer to the flag on page 262. SOLUTION:

A figure has reflectional symmetry if the figure can There are a total of four possible lines that go

be mapped onto itself by a reflection in a line. through the center and are lines of reflections. Thus, the flag has four lines of symmetry. The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

ANSWER: yes; 4

Two diagonal lines of reflection are possible.

17. Refer to page 262. SOLUTION:

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. There are a total of four possible lines that go through the center and are lines of reflections. Thus, the flag has four lines of symmetry. The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

It is not possible to reflect over a vertical or line through the diagonals.

ANSWER: Thus, the figure has one line of symmetry. yes; 4 ANSWER: yes; 1

The figure has reflectional symmetry. SOLUTION: A figure in the plane has rotational symmetry if the A horizontal line is a line of reflections for this flag. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

It is not possible to reflect over a vertical or line through the diagonals. The figure has rotational symmetry.

Thus, the figure has one line of symmetry. The number of times a figure maps onto itself as it ANSWER: rotates from 0° to 360° is called the order of symmetry. yes; 1 This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps State whether the figure has rotational onto itself. symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state The figure has a magnitude of symmetry of the order and magnitude of symmetry. .

ANSWER:

18. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation yes; 2; 180° between 0° and 360° about the center of the figure.

19.

SOLUTION: The figure has rotational symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The number of times a figure maps onto itself as it between 0° and 360° about the center of the figure. rotates from 0° to 360° is called the order of symmetry. The triangle has rotational symmetry.

This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps

onto itself. The number of times a figure maps onto itself as it The figure has a magnitude of symmetry of rotates from 0° to 360° is called the order of . symmetry.

ANSWER: The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

yes; 2; 180° The figure has magnitude of symmetry of .

ANSWER:

19. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation yes; 3; 120° between 0° and 360° about the center of the figure.

The triangle has rotational symmetry.

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation

between 0° and 360° about the center of the figure.

The number of times a figure maps onto itself as it The isosceles trapezoid has no rotational symmetry. rotates from 0° to 360° is called the order of There is no way to rotate it such that it can be symmetry. mapped onto itself.

The figure has order 3 rotational symmetry. ANSWER:

no The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of . 21.

ANSWER: SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The crescent shaped figure has no rotational yes; 3; 120° symmetry. There is no way to rotate it such that it can be mapped onto itself.

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

ANSWER: no

21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation ANSWER: between 0° and 360° about the center of the figure. no

The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

ANSWER: The magnitude of symmetry is the smallest angle no through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

22. ANSWER: SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry. yes; 8; 45°

23.

SOLUTION: A figure in the plane has rotational symmetry if the The number of times a figure maps onto itself as it figure can be mapped onto itself by a rotation rotates from 0° to 360° is called the order of between 0° and 360° about the center of the figure. symmetry. The figure has rotational symmetry. The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The figure has magnitude of symmetry of symmetry. . ANSWER: The figure has order 8 rotational symmetry. This means that the figure can be rotated 8 times and map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

yes; 8; 45° The figure has magnitude of symmetry of .

ANSWER:

23. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation yes; 8; 45° between 0° and 360° about the center of the figure. WHEELS State whether each wheel cover appears The figure has rotational symmetry. to have rotational symmetry. Write yes or no. If so, state the order and magnitude of symmetry. 24. Refer to page 263. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The wheel has rotational symmetry. symmetry. The number of times a figure maps onto itself as it The figure has order 8 rotational symmetry. This rotates from 0° to 360° is called the order of means that the figure can be rotated 8 times and map symmetry. onto itself within 360°. The wheel has order 5 rotational symmetry. There The magnitude of symmetry is the smallest angle are 5 large spokes and 5 small spokes. You can through which a figure can be rotated so that it maps rotate the wheel 5 times within 360° and map the onto itself. figure onto itself.

The figure has magnitude of symmetry of The magnitude of symmetry is the smallest angle . through which a figure can be rotated so that it maps onto itself. ANSWER: The wheel has magnitude of symmetry .

ANSWER: 3-5 Symmetry yes; 5; 72° yes; 8; 45°

WHEELS State whether each wheel cover appears 25. Refer to page 263. to have rotational symmetry. Write yes or no. If so, SOLUTION: state the order and magnitude of symmetry. 24. Refer to page 263. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The wheel has rotational symmetry. between 0° and 360° about the center of the figure. The number of times a figure maps onto itself as it The wheel has rotational symmetry. rotates from 0° to 360° is called the order of symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The wheel has order 8 rotational symmetry. There symmetry. are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself. The wheel has order 5 rotational symmetry. There are 5 large spokes and 5 small spokes. You can The magnitude of symmetry is the smallest angle rotate the wheel 5 times within 360° and map the through which a figure can be rotated so that it maps figure onto itself. onto itself.

The magnitude of symmetry is the smallest angle The wheel has order 8 rotational symmetry and through which a figure can be rotated so that it maps magnitude . onto itself. ANSWER: The wheel has magnitude of symmetry yes; 8; 45° . 26. Refer to page 263. ANSWER: SOLUTION: yes; 5; 72° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 25. Refer to page 263. between 0° and 360° about the center of the figure. The wheel has rotational symmetry. SOLUTION:

between 0° and 360° about the center of the figure. symmetry. The wheel has order 10 rotational symmetry. There are 10 bolts and the tire can be The wheel has rotational symmetry. rotated 10 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle State whether the figure has line symmetry through which a figure can be rotated so that it maps and/or rotational symmetry. If so, describe the onto itself. reflections and/or rotations that map the figure onto itself. eSolutionsThe Manualwheel -hasPowered orderby 8Cognero rotational symmetry and Page 11 magnitude .

ANSWER: yes; 8; 45° 26. Refer to page 263. 27. SOLUTION: A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation This triangle is scalene, so it cannot have symmetry. between 0° and 360° about the center of the figure. The wheel has rotational symmetry. ANSWER: no symmetry The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The wheel has order 10 rotational symmetry. There are 10 bolts and the tire can be rotated 10 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps 28. onto itself. The wheel has magnitude of symmetry of . SOLUTION: This figure is a square, because each pair of adjacent ANSWER: sides is congruent and perpendicular. yes; 10; 36° All squares have both line and rotational symmetry. The line symmetry is vertically, horizontally, and State whether the figure has line symmetry diagonally through the center of the square, with lines and/or rotational symmetry. If so, describe the that are either parallel to the sides of the square or reflections and/or rotations that map the figure that include two vertices of the square. The onto itself. equations of those lines are: x = 0, y = 0, y = x, and y = -x

The rotational symmetry is for each quarter turn in a square, so the rotations of 90, 180, and 270 degrees around the origin map the square onto itself.

27. ANSWER: line symmetry; rotational symmetry; the reflection in SOLUTION: the line x = 0, the reflection in the line y = 0, the This triangle is scalene, so it cannot have symmetry. reflection in the line y = x, and the reflection in the line y = -x all map the square onto itself; the rotations ANSWER: of 90, 180, and 270 degrees around the origin map no symmetry the square onto itself.

28. 29. SOLUTION: SOLUTION: This figure is a square, because each pair of adjacent The trapezoid has line symmetry, because it is sides is congruent and perpendicular. isosceles, but it does not have rotational symmetry, All squares have both line and rotational symmetry. because no trapezoid does. The line symmetry is vertically, horizontally, and diagonally through the center of the square, with lines The reflection in the line y = 1.5 maps the trapezoid that are either parallel to the sides of the square or onto itself, because that is the perpendicular bisector that include two vertices of the square. The to the parallel sides. equations of those lines are: x = 0, y = 0, y = x, and y = -x ANSWER: line symmetry; the reflection in the line y = 1.5 maps The rotational symmetry is for each quarter turn in a the trapezoid onto itself. square, so the rotations of 90, 180, and 270 degrees around the origin map the square onto itself.

ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = 0, the reflection in the line y = x, and the reflection in the 30. line y = -x all map the square onto itself; the rotations of 90, 180, and 270 degrees around the origin map SOLUTION: the square onto itself. This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its center, which is the point (1, -1.5).

Since this parallelogram is not a rhombus it does not have line symmetry.

ANSWER: 29. rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram SOLUTION: onto itself. The trapezoid has line symmetry, because it is isosceles, but it does not have rotational symmetry, 31. MODELING Symmetry is an important component because no trapezoid does. of photography. Photographers often use reflection in water to create symmetry in photos. The photo on The reflection in the line y = 1.5 maps the trapezoid page 263 is a long exposure shot of the Eiffel tower onto itself, because that is the perpendicular bisector reflected in a pool. to the parallel sides. a. Describe the two-dimensional symmetry created ANSWER: by the photo. line symmetry; the reflection in the line y = 1.5 maps b. Is there rotational symmetry in the photo? Explain the trapezoid onto itself. your reasoning. SOLUTION: a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is 30. reflected over the horizontal line, there is no rotational symmetry. SOLUTION: This figure is a parallelogram, so it has rotational ANSWER: symmetry of a half turn or 180 degrees around its a. Sample answer: There is a horizontal line of center, which is the point (1, -1.5). symmetry between the tower and its reflection. There is a vertical line of symmetry through the Since this parallelogram is not a rhombus it does not center of the photo. have line symmetry. b No; sample answer: Because of how the image is ANSWER: reflected over the horizontal line, there is no rotational symmetry. rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram COORDINATE GEOMETRY Determine

onto itself. whether the figure with the given vertices has line symmetry and/or rotational symmetry. 31. MODELING Symmetry is an important component 32. R(–3, 3), S(–3, –3), T(3, 3) of photography. Photographers often use reflection in water to create symmetry in photos. The photo on SOLUTION: page 263 is a long exposure shot of the Eiffel tower Draw the figure on a coordinate plane. reflected in a pool.

b No; sample answer: Because of how the image is reflected over the horizontal line, there is no The given triangle has a line of symmetry through rotational symmetry. points (0, 0) and (–3, 3).

ANSWER: A figure in the plane has rotational symmetry if the a. Sample answer: There is a horizontal line of figure can be mapped onto itself by a rotation symmetry between the tower and its reflection. between 0° and 360° about the center of the figure. There is a vertical line of symmetry through the There is not way to rotate the figure and have it map center of the photo. onto itself. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no Thus, the figure has only line symmetry. rotational symmetry. ANSWER: COORDINATE GEOMETRY Determine line whether the figure with the given vertices has line symmetry and/or rotational symmetry. 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) 32. R(–3, 3), S(–3, –3), T(3, 3) SOLUTION: SOLUTION: Draw the figure on a coordinate plane. Draw the figure on a coordinate plane.

A figure has line symmetry if the figure can be A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The

mapped onto itself by a reflection in a line. given figure has 4 lines of symmetry. The line of symmetry are though the following pairs of points The given triangle has a line of symmetry through {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, points (0, 0) and (–3, 3). and {(2, 2), (2, –2)}.

A figure in the plane has rotational symmetry if the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. between 0° and 360° about the center of the figure. There is not way to rotate the figure and have it map The figure can be rotated from the origin and map onto itself. onto itself. The order of symmetry is 4.

Thus, the figure has only line symmetry. Thus, the figure has both line symmetry and rotational symmetry. ANSWER: line ANSWER: line and rotational 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – SOLUTION: 2) Draw the figure on a coordinate plane. SOLUTION: Draw the figure on a coordinate plane.

A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The A figure has line symmetry if the figure can be given figure has 4 lines of symmetry. The line of mapped onto itself by a reflection in a line. The symmetry are though the following pairs of points given hexagon has 2 lines of symmetry. The lines {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, pass through the following pair of points {(0, 4), (0, – and {(2, 2), (2, –2)}. 4)}, and {(3, 0), (–3, 0)}

A figure in the plane has rotational symmetry if the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map The figure has rotational symmetry. You can rotate onto itself. The order of symmetry is 4. the figure once within 360° and have it map to itself.

Thus, the figure has both line symmetry and Thus, the figure has both line symmetry and rotational symmetry. rotational symmetry.

ANSWER: ANSWER: line and rotational line and rotational 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) 2) SOLUTION: SOLUTION: Draw the figure on a coordinate plane. Draw the figure on a coordinate plane.

A figure has line symmetry if the figure can be A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The mapped onto itself by a reflection in a line. The given hexagon has 2 lines of symmetry. The lines trapezoid has a line of reflection through points (0,3) pass through the following pair of points {(0, 4), (0, – and (0, –3). 4)}, and {(3, 0), (–3, 0)} A figure in the plane has rotational symmetry if the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure can be mapped onto itself by a rotation between 0° and 360° about the center of the between 0° and 360° about the center of the figure. figure. There is no way to rotate this figure and have The figure has rotational symmetry. You can rotate it map onto itself. Thus, it does not have rotational the figure once within 360° and have it map to itself. symmetry.

Thus, the figure has both line symmetry and Therefore, the figure has only line symmetry. rotational symmetry. ANSWER: ANSWER: line line and rotational ALGEBRA Graph the function and determine 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) whether the graph has line and/or rotational symmetry. If so, state the order and magnitude of SOLUTION: symmetry, and write the equations of any lines of Draw the figure on a coordinate plane. symmetry. 36. y = x SOLUTION: Graph the function.

A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The trapezoid has a line of reflection through points (0,3)

and (0, –3).

A figure has reflectional symmetry if the figure can A figure in the plane has rotational symmetry if the be mapped onto itself by a reflection in a line. The figure can be mapped onto itself by a rotation between 0° and 360° about the center of the line y = x has reflectional symmetry since any line figure. There is no way to rotate this figure and have perpendicular to y = x is a line of reflection. The it map onto itself. Thus, it does not have rotational equation of the line symmetry is y = –x. symmetry. A figure in the plane has rotational symmetry if the Therefore, the figure has only line symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. ANSWER: The line can be rotated twice within 360° and be line mapped onto itself.

ALGEBRA Graph the function and determine The number of times a figure maps onto itself as it whether the graph has line and/or rotational rotates from 0° to 360° is called the order of symmetry. If so, state the order and magnitude of symmetry. The graph has order 2 rotational symmetry, and write the equations of any lines of symmetry. symmetry. 36. y = x The magnitude of symmetry is the smallest angle SOLUTION: through which a figure can be rotated so that it maps onto itself. Graph the function. The graph has magnitude of symmetry of .

Thus, the graph has both reflectional and rotational symmetry.

ANSWER: rotational; 2; 180°; line symmetry; y = –x

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 2 between 0° and 360° about the center of the figure. 37. y = x + 1 The line can be rotated twice within 360° and be SOLUTION: mapped onto itself. Graph the function. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of . A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The Thus, the graph has both reflectional and rotational graph is reflected through the y-axis. Thus, the symmetry. equation of the line symmetry is x = 0. ANSWER: rotational; 2; 180°; line symmetry; y = –x A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. There is no way to rotate the graph and have it map onto itself.

Thus, the graph has only reflectional symmetry.

ANSWER: line; x = 0

2 37. y = x + 1 SOLUTION: Graph the function.

38. y = –x3 SOLUTION: Graph the function.

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph is reflected through the y-axis. Thus, the equation of the line symmetry is x = 0.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. There is no way to rotate the graph and have A figure has reflectional symmetry if the figure can it map onto itself. be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the Thus, the graph has only reflectional symmetry. graph can be mapped onto itself.

ANSWER: A figure in the plane has rotational symmetry if the line; x = 0 figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. You can rotate the graph through the origin and have it map onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry.

The magnitude of symmetry is the smallest angle 38. y = –x3 through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of SOLUTION: . Graph the function. Thus, the graph has only rotational symmetry.

ANSWER: rotational; 2; 180°

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself. 39. Refer to the rectangle on the coordinate plane. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. You can rotate the graph through the origin and have it map onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational a. What are the equations of the lines of symmetry of symmetry. the rectangle? b. What happens to the equations of the lines of The magnitude of symmetry is the smallest angle symmetry when the rectangle is rotated 90 degrees through which a figure can be rotated so that it maps counterclockwise around its center of symmetry? onto itself. The graph has magnitude of symmetry of Explain. . SOLUTION: Thus, the graph has only rotational symmetry. a. The lines of symmetry are parallel to the sides of the rectangles, and through the center of rotation. ANSWER: rotational; 2; 180° The slopes of the sides of the rectangle are 0.5 and -2, so the slopes of the lines of symmetry are the same. The center of the rectangle is (1, 1.5). Use the point-slope formula to find equations.

b. The equations of the lines of symmetry do not change; although the rectangle does not map onto 39. Refer to the rectangle on the coordinate plane. itself under this rotation, the lines of symmetry are mapped to each other.

ANSWER: a. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are a. What are the equations of the lines of symmetry of mapped to each other. the rectangle? b. What happens to the equations of the lines of 40. MULTIPLE REPRESENTATIONS In this symmetry when the rectangle is rotated 90 degrees problem, you will use dynamic geometric software to counterclockwise around its center of symmetry? investigate line and rotational symmetry in regular polygons. Explain.

SOLUTION: a. Geometric Use The Geometer’s Sketchpad to a. The lines of symmetry are parallel to the sides of draw an equilateral triangle. Use the reflection tool the rectangles, and through the center of rotation. under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. The slopes of the sides of the rectangle are 0.5 and b. Geometric Use the rotation tool under the -2, so the slopes of the lines of symmetry are the transformation menu to investigate the rotational same. symmetry of the figure in part a. Then record its The center of the rectangle is (1, 1.5). Use the order of symmetry. point-slope formula to find equations. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of b. The equations of the lines of symmetry do not lines of symmetry and the order of symmetry for a change; although the rectangle does not map onto regular polygon with n sides. itself under this rotation, the lines of symmetry are SOLUTION: mapped to each other. a. Construct an equilateral triangle and label the ANSWER: vertices A, B, and C. Draw a line through A a. perpendicular to . Reflect the triangle in the line. b. The equations of the lines of symmetry do not Show the labels of the reflected image. If the image change; although the rectangle does not map onto maps to the original, then this line is a line of reflection. itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and

determine all possible lines of symmetry. Then record Next, draw a line through B perpendicular to . their number. Reflect the triangle in the line. Show the labels of the b. Geometric Use the rotation tool under the reflected image. If the image maps to the original, transformation menu to investigate the rotational then this line is a line of reflection. symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides.

SOLUTION: Lastly, draw a line through C perpendicular to . a. Construct an equilateral triangle and label the Reflect the triangle in the line. Show the labels of the vertices A, B, and C. Draw a line through A reflected image. If the image maps to the original, perpendicular to . Reflect the triangle in the line. then this line is a line of reflection. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection. There are 3 lines of symmetry.

labels of the image. Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

There are 3 lines of symmetry.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Since the figure maps onto itself 3 times as it is Show the labels of the image rotated, the order of symmetry is 3. c. Square Construct a square and then construct lines through the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines Since the figure maps onto itself 3 times as it is constructed. So there are 5 lines of symmetry. rotated, the order of symmetry is 3. Next, rotate the square about the center point. The c. image maps to the original at 72, 144, 216, 288, and Square 360 degree rotations. So the order of symmetry is 5. Construct a square and then construct lines through the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Regular Hexagon Construct a regular hexagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 6 lines constructed. So there are 6 lines of symmetry. Next, rotate the square about the center point. The Regular Pentagon image maps to the original at 60, 120, 180, 240, 300, Construct a regular pentagon and then construct lines and 360 degree rotations. So the order of symmetry through each vertex perpendicular to the sides. Use is 6. the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5.

d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and Regular Hexagon order of symmetry n. Construct a regular hexagon and then construct lines through each vertex perpendicular to the sides. Use ANSWER: the reflection tool first to find that the image maps a. 3 onto the original when reflected in each of the 6 lines b. 3 constructed. So there are 6 lines of symmetry. c. Next, rotate the square about the center point. The image maps to the original at 60, 120, 180, 240, 300, and 360 degree rotations. So the order of symmetry is 6. d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. This figure has 4 lines of symmetry.

ANSWER:

a. 3 A figure in the plane has rotational symmetry if the b. 3 figure can be mapped onto itself by a rotation c. between 0° and 360° about the center of the figure.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has The figure also has rotational symmetry. only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them Therefore, neither of them are correct. Figure A has correct? Explain your reasoning. both line and rotational symmetry.

ANSWER: Neither; Figure A has both line and rotational symmetry.

42. CHALLENGE A quadrilateral in the coordinate SOLUTION: plane has exactly two lines of symmetry, y = x – 1 A figure has line symmetry if the figure can be and y = –x + 2. Find a set of possible vertices for mapped onto itself by a reflection in a line. This the figure. Graph the figure and the lines of figure has 4 lines of symmetry. symmetry. SOLUTION: Graph the figure and the lines of symmetry.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Pick points that are the same distance a from one line and the same distance b from the other line. In the same answer, the quadrilateral is a rectangle with sides which are parallel to the lines of symmetry. This guarantees that the vertices of the quadrilateral The figure also has rotational symmetry. are the same distance a from one line and the same distance b from the other line. In this case, a = Therefore, neither of them are correct. Figure A has both line and rotational symmetry. and b = .

43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. SOLUTION: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such Pick points that are the same distance a from one lines. line and the same distance b from the other line. In the same answer, the quadrilateral is a rectangle with ANSWER: sides which are parallel to the lines of symmetry. circle; Every line through the center of a circle is a This guarantees that the vertices of the quadrilateral line of symmetry, and there are infinitely many such are the same distance a from one line and the same lines. distance b from the other line. In this case, a = 44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. and b = . SOLUTION: A set of possible vertices for the figure are, (–1, 0), A figure has line symmetry if the figure can be (2, 3), (4, 1), and (1, 2). mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can ANSWER: be mapped onto itself by a rotation between 0° and Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) 360° about the center of the figure.

Identify a figure that has line symmetry but does not have rotational symmetry.

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain.

SOLUTION: circle; Every line through the center of a circle is a ANSWER: line of symmetry, and there are infinitely many such Sample answer: An isosceles triangle has line lines. symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry ANSWER: because it cannot be rotated from 0° to 360° and circle; Every line through the center of a circle is a map onto itself. line of symmetry, and there are infinitely many such lines.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain.

SOLUTION: WRITING IN MATH A figure has line symmetry if the figure can be 45. How are line symmetry and mapped onto itself by a reflection in a line. A figure rotational symmetry related? in the plane has rotational symmetry if the figure can SOLUTION: be mapped onto itself by a rotation between 0° and In both types of symmetries the figure is mapped 360° about the center of the figure. onto itself.

Identify a figure that has line symmetry but does not Rotational symmetry. have rotational symmetry.

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does Reflectional symmetry: not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

In some cases an object can have both rotational and reflectional symmetry, such as the diamond, however ANSWER: some objects do not have both such as the crab. Sample answer: An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

ANSWER:

Sample answer: In both rotational and line symmetry 45. WRITING IN MATH How are line symmetry and a figure is mapped onto itself. However, in line rotational symmetry related? symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is SOLUTION: mapped onto itself by a rotation. A figure can have In both types of symmetries the figure is mapped line symmetry and rotational symmetry. onto itself. 46. Sasha owns a tile store. For each tile in her store, she Rotational symmetry. calculates the sum of the number of lines of symmetry and the order of symmetry, and then she enters this value into a database. Which value should she enter in the database for the tile shown here? Reflectional symmetry:

In some cases an object can have both rotational and A 2 reflectional symmetry, such as the diamond, however B 3 some objects do not have both such as the crab. C 4 D 8 SOLUTION:

ANSWER: The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile. Sample answer: In both rotational and line symmetry

a figure is mapped onto itself. However, in line It has an order of symmetry of 2, because it has symmetry the figure is mapped onto itself by a rotational symmetry at 180 degrees, or each half turn reflection, and in rotational symmetry, a figure is around its center. mapped onto itself by a rotation. A figure can have

line symmetry and rotational symmetry. 2 + 2 = 4, so C is the correct answer.

46. Sasha owns a tile store. For each tile in her store, she ANSWER: calculates the sum of the number of lines of symmetry and the order of symmetry, and then she C enters this value into a database. Which value should 47. Patrick drew a figure that has rotational symmetry she enter in the database for the tile shown here? but not line symmetry. Which of the following could be the figure that Patrick drew? A

A 2 B 3 C 4 C D 8 SOLUTION: D

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile. SOLUTION:

It has an order of symmetry of 2, because it has Option A has rotational and reflectional symmetry. rotational symmetry at 180 degrees, or each half turn around its center.

2 + 2 = 4, so C is the correct answer.

Option B has reflectional symmetry but not rotational ANSWER: symmetry. C

Option D has rotational symmetry but not reflectional C symmetry.

Option E has reflectional symmetry but not rotational

E symmetry.

SOLUTION: Option A has rotational and reflectional symmetry.

The correct choice is D.

Option B has reflectional symmetry but not rotational ANSWER: symmetry. D

48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle Option C has neither rotational nor reflectional D Scalene triangle symmetry. SOLUTION: An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

Option D has rotational symmetry but not reflectional symmetry. ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS Option E has reflectional symmetry but not rotational has line symmetry but not rotational symmetry? symmetry. A B C D SOLUTION: First, plot the points.

The correct choice is D.

ANSWER: D

48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle D Scalene triangle SOLUTION:

An isosceles triangle has one line of symmetry and Then, plot each option A-D to consider each figure no rotational symmetry. The correct choice is C. and its symmetry. Option A has both reflectional and rotational symmetry. ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D SOLUTION: First, plot the points. Option B has reflective symmetry but not rotational symmetry. The correct choice is B.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

Thus, there are four possible lines that go through the center and are lines of reflections.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of 2. reflection must go through the center point. SOLUTION: A figure has reflectional symmetry if the figure can Two lines of reflection go through the sides of the be mapped onto itself by a reflection in a line. figure. The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER: no

Two lines of reflection go through the vertices of the figure.

3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry. Thus, there are four possible lines that go through the center and are lines of reflections. The figure has a vertical line of symmetry.

It does not have a horizontal line of symmetry.

Therefore, the figure has four lines of symmetry.

ANSWER:

yes; 4 The figure does not have a line of symmetry through the vertices.

Thus, the figure has only one line of symmetry.

2. ANSWER: SOLUTION: yes; 1 A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself. ANSWER: State whether the figure has rotational no symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

3. SOLUTION: 4. A figure has reflectional symmetry if the figure can SOLUTION: be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the

figure can be mapped onto itself by a rotation The given figure has reflectional symmetry. between 0° and 360° about the center of the figure.

The figure has a vertical line of symmetry. For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: no

It does not have a horizontal line of symmetry.

5. SOLUTION: The figure does not have a line of symmetry through A figure in the plane has rotational symmetry if the the vertices. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

Thus, the figure has only one line of symmetry.

ANSWER: yes; 1

The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of symmetry.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is . 4. ANSWER: SOLUTION: yes; 2; 180° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: no 6. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. 5. The given figure has rotational symmetry. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

The magnitude of symmetry is the smallest angle The number of times a figure maps onto itself as it through which a figure can be rotated so that it maps rotates form 0° and 360° is called the order of onto itself. symmetry. The figure has magnitude of symmetry of The given figure has order of symmetry of 2, since . the figure can be rotated twice in 360°. ANSWER: The magnitude of symmetry is the smallest angle yes; 4; 90° through which a figure can be rotated so that it maps onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is .

ANSWER: yes; 2; 180° State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

7. 6. SOLUTION: SOLUTION: Vertical and horizontal lines through the center and A figure in the plane has rotational symmetry if the diagonal lines through two opposite vertices are all figure can be mapped onto itself by a rotation lines of symmetry for a square oriented this way. between 0° and 360° about the center of the figure. The equations of those lines in this figure are x = 0, The given figure has rotational symmetry. y = -1, y = x - 1, and y = -x - 1.

Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

ANSWER:

line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the The number of times a figure maps onto itself as it reflection in the line y = x - 1, and the reflection in rotates from 0° to 360° is called the order of the line y = -x - 1 map the square onto itself; the symmetry. rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself. Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of . 8. ANSWER: SOLUTION: yes; 4; 90° This figure does not have line symmetry, because adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

ANSWER: rotational symmetry; the rotation of 180 degrees State whether the figure has line symmetry around the point (1, 1) maps the parallelogram onto and/or rotational symmetry. If so, describe the itself. reflections and/or rotations that map the figure onto itself. REGULARITY State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

7. 9. SOLUTION: SOLUTION: Vertical and horizontal lines through the center and A figure has reflectional symmetry if the figure can diagonal lines through two opposite vertices are all be mapped onto itself by a reflection in a line. lines of symmetry for a square oriented this way. For the given figure, there are no lines of reflection The equations of those lines in this figure are x = 0, where the figure can map onto itself. Thus, the figure y = -1, y = x - 1, and y = -x - 1. does not have any lines of of symmetry.

Each quarter turn also maps the square onto itself. ANSWER: So the rotations of 90, 180, and 270 degrees around no the point (0, -1) map the square onto itself.

ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the 10. rotations of 90, 180, and 270 degrees around the point SOLUTION: (0, -1) map the square onto itself. A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

8. The figure has a vertical and horizontal line of reflection. SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself. ANSWER: It is also possible to have reflection over the rotational symmetry; the rotation of 180 degrees diagonal lines. around the point (1, 1) maps the parallelogram onto itself.

9. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection ANSWER: where the figure can map onto itself. Thus, the figure yes; 4 does not have any lines of of symmetry.

ANSWER: no

10. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line. 11.

SOLUTION: The given figure has reflectional symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. In order for the figure to map onto itself, the line of reflection must go through the center point. The given hexagon has reflectional symmetry. The figure has a vertical and horizontal line of reflection. In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though opposites edges.

It is also possible to have reflection over the diagonal lines.

There are three lines of reflection that go though opposites vertices.

Therefore, the figure has four lines of symmetry

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry.

ANSWER: yes; 4

ANSWER: yes; 6

11. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. 12. The given hexagon has reflectional symmetry. SOLUTION: A figure has reflectional symmetry if the figure can In order for the hexagon to map onto itself, the line be mapped onto itself by a reflection in a line. of reflection must go through the center point.

The figure has reflectional symmetry. There are three lines of reflection that go though

opposites edges. There is only one line of symmetry, a horizontal line through the middle of the figure.

Thus, the figure has one line of symmetry. There are three lines of reflection that go though opposites vertices. ANSWER: yes; 1

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has 13. six lines of symmetry. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

There is only one possible line of reflection, horizontally though the middle of the figure. ANSWER: yes; 6

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

12. 14. SOLUTION: SOLUTION: A figure has reflectional symmetry if the figure can A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line. be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. The given figure does not have reflectional symmetry. It is not possible to draw a line of There is only one line of symmetry, a horizontal line reflection where the figure can map onto itself. through the middle of the figure. ANSWER: no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the Thus, the figure has one line of symmetry. flag, draw all lines of symmetry, and state their number. ANSWER: 15. Refer to page 262. yes; 1 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, 13. the flag would have three lines of symmetry. SOLUTION: ANSWER: A figure has reflectional symmetry if the figure can no be mapped onto itself by a reflection in a line. 16. Refer to the flag on page 262. The figure has reflectional symmetry. SOLUTION: There is only one possible line of reflection, A figure has reflectional symmetry if the figure can horizontally though the middle of the figure. be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of Thus, the figure has one line of symmetry. reflection must go through the center point.

ANSWER: A horizontal and vertical lines of reflection are yes; 1 possible.

14. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. Two diagonal lines of reflection are possible.

The given figure does not have reflectional symmetry. It is not possible to draw a line of reflection where the figure can map onto itself.

ANSWER: no

FLAGS State whether each flag design appears to

have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their There are a total of four possible lines that go number. through the center and are lines of reflections. Thus, 15. Refer to page 262. the flag has four lines of symmetry. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER:

no ANSWER: 16. Refer to the flag on page 262. yes; 4 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible. 17. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

Two diagonal lines of reflection are possible.

It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

There are a total of four possible lines that go through the center and are lines of reflections. Thus, the flag has four lines of symmetry.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

18. ANSWER: SOLUTION: yes; 4 A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

17. Refer to page 262. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of SOLUTION: symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto The figure has reflectional symmetry. itself.

A horizontal line is a line of reflections for this flag. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has a magnitude of symmetry of .

It is not possible to reflect over a vertical or line ANSWER: through the diagonals.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1 yes; 2; 180°

19.

SOLUTION: State whether the figure has rotational symmetry. Write yes or no. If so, copy the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure, locate the center of symmetry, and state between 0° and 360° about the center of the figure. the order and magnitude of symmetry.

The triangle has rotational symmetry.

18. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 3 rotational symmetry. The figure has rotational symmetry. The magnitude of symmetry is the smallest angle The number of times a figure maps onto itself as it through which a figure can be rotated so that it maps rotates from 0° to 360° is called the order of onto itself. symmetry. The figure has magnitude of symmetry of This figure has order 2 rotational symmetry, since . you have to rotate 180° to get the figure to map onto itself. ANSWER:

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has a magnitude of symmetry of yes; 3; 120° .

ANSWER:

20. SOLUTION: A figure in the plane has rotational symmetry if the yes; 2; 180° figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

19. SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation no between 0° and 360° about the center of the figure.

The triangle has rotational symmetry.

21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 3; 120°

20. SOLUTION: A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation no between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself. 22.

ANSWER: SOLUTION: no A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 8; 45°

ANSWER: no 23. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. 22. The figure has rotational symmetry. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This means that the figure can be rotated 8 times and map onto itself within 360°.

The figure has order 8 rotational symmetry. This The figure has magnitude of symmetry of implies you can rotate the figure 8 times and have it . map onto itself within 360°. ANSWER: The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of . yes; 8; 45°

The wheel has rotational symmetry.

The number of times a figure maps onto itself as it 23. rotates from 0° to 360° is called the order of symmetry. SOLUTION: A figure in the plane has rotational symmetry if the The wheel has order 5 rotational symmetry. There figure can be mapped onto itself by a rotation are 5 large spokes and 5 small spokes. You can between 0° and 360° about the center of the figure. rotate the wheel 5 times within 360° and map the figure onto itself. The figure has rotational symmetry. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The wheel has magnitude of symmetry .

The number of times a figure maps onto itself as it ANSWER: rotates from 0° to 360° is called the order of symmetry. yes; 5; 72°

The figure has order 8 rotational symmetry. This 25. Refer to page 263. means that the figure can be rotated 8 times and map SOLUTION: onto itself within 360°. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The magnitude of symmetry is the smallest angle between 0° and 360° about the center of the figure. through which a figure can be rotated so that it maps onto itself. The wheel has rotational symmetry. The figure has magnitude of symmetry of . The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of ANSWER: symmetry.

The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself.

yes; 8; 45° The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps WHEELS State whether each wheel cover appears onto itself. to have rotational symmetry. Write yes or no. If so, state the order and magnitude of symmetry. The wheel has order 8 rotational symmetry and 24. Refer to page 263. magnitude . SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation yes; 8; 45° between 0° and 360° about the center of the figure. 26. Refer to page 263.

The wheel has rotational symmetry. SOLUTION: A figure in the plane has rotational symmetry if the The number of times a figure maps onto itself as it figure can be mapped onto itself by a rotation rotates from 0° to 360° is called the order of between 0° and 360° about the center of the figure. symmetry. The wheel has rotational symmetry.

The wheel has order 5 rotational symmetry. There The number of times a figure maps onto itself as it are 5 large spokes and 5 small spokes. You can rotates from 0° to 360° is called the order of rotate the wheel 5 times within 360° and map the symmetry. The wheel has order 10 rotational figure onto itself. symmetry. There are 10 bolts and the tire can be rotated 10 times within 360° and map onto itself. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The magnitude of symmetry is the smallest angle onto itself. through which a figure can be rotated so that it maps onto itself. The wheel has magnitude of symmetry of The wheel has magnitude of symmetry . . ANSWER: ANSWER: yes; 10; 36° yes; 5; 72° State whether the figure has line symmetry and/or rotational symmetry. If so, describe the 25. Refer to page 263. reflections and/or rotations that map the figure SOLUTION: onto itself. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The wheel has rotational symmetry.

The number of times a figure maps onto itself as it 27. rotates from 0° to 360° is called the order of symmetry. SOLUTION: This triangle is scalene, so it cannot have symmetry. The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times ANSWER: within 360° and map onto itself. no symmetry

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The wheel has order 8 rotational symmetry and magnitude . 28. ANSWER: yes; 8; 45° SOLUTION: This figure is a square, because each pair of adjacent 26. Refer to page 263. sides is congruent and perpendicular. All squares have both line and rotational symmetry. SOLUTION: The line symmetry is vertically, horizontally, and A figure in the plane has rotational symmetry if the diagonally through the center of the square, with lines figure can be mapped onto itself by a rotation that are either parallel to the sides of the square or between 0° and 360° about the center of the figure. that include two vertices of the square. The The wheel has rotational symmetry. equations of those lines are: x = 0, y = 0, y = x, and y = -x The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The rotational symmetry is for each quarter turn in a symmetry. The wheel has order 10 rotational square, so the rotations of 90, 180, and 270 degrees symmetry. There are 10 bolts and the tire can be around the origin map the square onto itself. rotated 10 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps line symmetry; rotational symmetry; the reflection in onto itself. The wheel has magnitude of symmetry of the line x = 0, the reflection in the line y = 0, the . reflection in the line y = x, and the reflection in the line y = -x all map the square onto itself; the rotations 3-5 SymmetryANSWER: of 90, 180, and 270 degrees around the origin map yes; 10; 36° the square onto itself. State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

29. SOLUTION: The trapezoid has line symmetry, because it is isosceles, but it does not have rotational symmetry, 27. because no trapezoid does.

SOLUTION: The reflection in the line y = 1.5 maps the trapezoid This triangle is scalene, so it cannot have symmetry. onto itself, because that is the perpendicular bisector to the parallel sides. ANSWER: no symmetry ANSWER: line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself.

28. SOLUTION: 30. This figure is a square, because each pair of adjacent sides is congruent and perpendicular. SOLUTION: All squares have both line and rotational symmetry. This figure is a parallelogram, so it has rotational The line symmetry is vertically, horizontally, and symmetry of a half turn or 180 degrees around its diagonally through the center of the square, with lines center, which is the point (1, -1.5). that are either parallel to the sides of the square or that include two vertices of the square. The Since this parallelogram is not a rhombus it does not equations of those lines are: x = 0, y = 0, y = x, and y have line symmetry. = -x ANSWER: The rotational symmetry is for each quarter turn in a rotational symmetry; the rotation of 180 degrees square, so the rotations of 90, 180, and 270 degrees around the point (1, -1.5) maps the parallelogram around the origin map the square onto itself. onto itself.

31. MODELING Symmetry is an important component ANSWER: of photography. Photographers often use reflection in line symmetry; rotational symmetry; the reflection in water to create symmetry in photos. The photo on the line x = 0, the reflection in the line y = 0, the page 263 is a long exposure shot of the Eiffel tower reflection in the line y = x, and the reflection in the reflected in a pool. line y = -x all map the square onto itself; the rotations of 90, 180, and 270 degrees around the origin map a. Describe the two-dimensional symmetry created the square onto itself. by the photo. b. Is there rotational symmetry in the photo? Explain your reasoning. eSolutions Manual - Powered by Cognero SOLUTION: Page 12 a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the 29. center of the photo. SOLUTION: b No; sample answer: Because of how the image is reflected over the horizontal line, there is no The trapezoid has line symmetry, because it is rotational symmetry. isosceles, but it does not have rotational symmetry, because no trapezoid does. ANSWER:

a. The reflection in the line y = 1.5 maps the trapezoid Sample answer: There is a horizontal line of onto itself, because that is the perpendicular bisector symmetry between the tower and its reflection. to the parallel sides. There is a vertical line of symmetry through the center of the photo. ANSWER: b No; sample answer: Because of how the image is line symmetry; the reflection in the line y = 1.5 maps reflected over the horizontal line, there is no the trapezoid onto itself. rotational symmetry. COORDINATE GEOMETRY Determine whether the figure with the given vertices has line symmetry and/or rotational symmetry. 32. R(–3, 3), S(–3, –3), T(3, 3) SOLUTION: Draw the figure on a coordinate plane. 30. SOLUTION: This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its center, which is the point (1, -1.5).

Since this parallelogram is not a rhombus it does not have line symmetry.

ANSWER: A figure has line symmetry if the figure can be rotational symmetry; the rotation of 180 degrees mapped onto itself by a reflection in a line. around the point (1, -1.5) maps the parallelogram onto itself. The given triangle has a line of symmetry through points (0, 0) and (–3, 3). 31. MODELING Symmetry is an important component of photography. Photographers often use reflection in A figure in the plane has rotational symmetry if the water to create symmetry in photos. The photo on figure can be mapped onto itself by a rotation page 263 is a long exposure shot of the Eiffel tower between 0° and 360° about the center of the figure. reflected in a pool. There is not way to rotate the figure and have it map onto itself. a. Describe the two-dimensional symmetry created by the photo. Thus, the figure has only line symmetry. b. Is there rotational symmetry in the photo? Explain your reasoning. ANSWER: SOLUTION: line a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) There is a vertical line of symmetry through the SOLUTION: center of the photo. Draw the figure on a coordinate plane. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no rotational symmetry.

ANSWER: a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no A figure has line symmetry if the figure can be rotational symmetry. mapped onto itself by a reflection in a line. The given figure has 4 lines of symmetry. The line of COORDINATE GEOMETRY Determine symmetry are though the following pairs of points whether the figure with the given vertices has line {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, symmetry and/or rotational symmetry. and {(2, 2), (2, –2)}. 32. R(–3, 3), S(–3, –3), T(3, 3) A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation Draw the figure on a coordinate plane. between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational

A figure has line symmetry if the figure can be 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – mapped onto itself by a reflection in a line. 2) SOLUTION: The given triangle has a line of symmetry through Draw the figure on a coordinate plane. points (0, 0) and (–3, 3).

Thus, the figure has only line symmetry. A figure has line symmetry if the figure can be ANSWER: mapped onto itself by a reflection in a line. The line given hexagon has 2 lines of symmetry. The lines pass through the following pair of points {(0, 4), (0, – 4)}, and {(3, 0), (–3, 0)} 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) SOLUTION: A figure in the plane has rotational symmetry if the Draw the figure on a coordinate plane. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure has rotational symmetry. You can rotate the figure once within 360° and have it map to itself.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER:

line and rotational A figure has line symmetry if the figure can be 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) mapped onto itself by a reflection in a line. The given figure has 4 lines of symmetry. The line of SOLUTION: symmetry are though the following pairs of points Draw the figure on a coordinate plane. {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, and {(2, 2), (2, –2)}.

Thus, the figure has both symmetry and line A figure has line symmetry if the figure can be rotational symmetry. mapped onto itself by a reflection in a line. The trapezoid has a line of reflection through points (0,3) ANSWER: and (0, –3). line and rotational 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – A figure in the plane has rotational symmetry if the 2) figure can be mapped onto itself by a rotation between 0° and 360° about the center of the SOLUTION: figure. There is no way to rotate this figure and have Draw the figure on a coordinate plane. it map onto itself. Thus, it does not have rotational symmetry.

Therefore, the figure has only line symmetry.

ANSWER: line

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational A figure has reflectional symmetry if the figure can 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) be mapped onto itself by a reflection in a line. The SOLUTION: line y = x has reflectional symmetry since any line Draw the figure on a coordinate plane. perpendicular to y = x is a line of reflection. The equation of the line symmetry is y = –x.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of A figure has line symmetry if the figure can be symmetry. The graph has order 2 rotational mapped onto itself by a reflection in a line. The symmetry. trapezoid has a line of reflection through points (0,3) and (0, –3). The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps A figure in the plane has rotational symmetry if the onto itself. figure can be mapped onto itself by a rotation The graph has magnitude of symmetry of between 0° and 360° about the center of the . figure. There is no way to rotate this figure and have

it map onto itself. Thus, it does not have rotational symmetry. Thus, the graph has both reflectional and rotational symmetry. Therefore, the figure has only line symmetry. ANSWER: ANSWER: rotational; 2; 180°; line symmetry; y = –x line

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The

line y = x has reflectional symmetry since any line perpendicular to y = x is a line of reflection. The A figure has reflectional symmetry if the figure can equation of the line symmetry is y = –x. be mapped onto itself by a reflection in a line. The graph is reflected through the y-axis. Thus, the A figure in the plane has rotational symmetry if the equation of the line symmetry is x = 0. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. A figure in the plane has rotational symmetry if the The line can be rotated twice within 360° and be figure can be mapped onto itself by a rotation mapped onto itself. between 0° and 360° about the center of the figure. There is no way to rotate the graph and have The number of times a figure maps onto itself as it it map onto itself. rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational Thus, the graph has only symmetry. symmetry. reflectional ANSWER: The magnitude of symmetry is the smallest angle line; x = 0 through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of .

Thus, the graph has both reflectional and rotational symmetry.

ANSWER:

rotational; 2; 180°; line symmetry; y = –x 38. y = –x3 SOLUTION: Graph the function.

2 37. y = x + 1 SOLUTION: Graph the function. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. You can rotate the graph through the origin

and have it map onto itself.

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of graph is reflected through the y-axis. Thus, the The graph has order 2 rotational equation of the line symmetry is x = 0. symmetry. symmetry.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The magnitude of symmetry is the smallest angle between 0° and 360° about the center of the through which a figure can be rotated so that it maps figure. There is no way to rotate the graph and have onto itself. The graph has magnitude of symmetry of it map onto itself. .

Thus, the graph has only reflectional symmetry. Thus, the graph has only rotational symmetry.

ANSWER: ANSWER: line; x = 0 rotational; 2; 180°

38. y = –x3 39. Refer to the rectangle on the coordinate plane. SOLUTION: Graph the function.

a. What are the equations of the lines of symmetry of the rectangle? b. What happens to the equations of the lines of symmetry when the rectangle is rotated 90 degrees counterclockwise around its center of symmetry? A figure has reflectional symmetry if the figure can Explain. be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the SOLUTION: graph can be mapped onto itself. a. The lines of symmetry are parallel to the sides of the rectangles, and through the center of rotation. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The slopes of the sides of the rectangle are 0.5 and between 0° and 360° about the center of the -2, so the slopes of the lines of symmetry are the figure. You can rotate the graph through the origin and have it map onto itself. same. The center of the rectangle is (1, 1.5). Use the The number of times a figure maps onto itself as it point-slope formula to find equations.

rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry. b. The equations of the lines of symmetry do not The magnitude of symmetry is the smallest angle change; although the rectangle does not map onto through which a figure can be rotated so that it maps itself under this rotation, the lines of symmetry are The graph has magnitude onto itself. of symmetry of mapped to each other. . ANSWER: Thus, the graph has only rotational symmetry. a. b. The equations of the lines of symmetry do not ANSWER: change; although the rectangle does not map onto rotational; 2; 180° itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and 39. Refer to the rectangle on the coordinate plane. determine all possible lines of symmetry. Then record their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the a. What are the equations of the lines of symmetry of order of symmetry for each polygon. the rectangle? d. Verbal Make a conjecture about the number of b. What happens to the equations of the lines of lines of symmetry and the order of symmetry for a symmetry when the rectangle is rotated 90 degrees regular polygon with n sides. counterclockwise around its center of symmetry? SOLUTION: Explain. a. Construct an equilateral triangle and label the SOLUTION: vertices A, B, and C. Draw a line through A a. The lines of symmetry are parallel to the sides of perpendicular to . Reflect the triangle in the line. the rectangles, and through the center of rotation. Show the labels of the reflected image. If the image maps to the original, then this line is a line of The slopes of the sides of the rectangle are 0.5 and reflection. -2, so the slopes of the lines of symmetry are the same. The center of the rectangle is (1, 1.5). Use the point-slope formula to find equations.

b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are Next, draw a line through B perpendicular to . mapped to each other. Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, ANSWER: then this line is a line of reflection. a. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons. Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the a. Geometric Use The Geometer’s Sketchpad to reflected image. If the image maps to the original, draw an equilateral triangle. Use the reflection tool then this line is a line of reflection. under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides. SOLUTION: a. Construct an equilateral triangle and label the vertices A, B, and C. Draw a line through A There are 3 lines of symmetry. perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image b. Construct an equilateral triangle and show the maps to the original, then this line is a line of labels of the vertices. Next, find the center of the reflection. triangle. Since this is an equilateral triangle, the circumcenter, incenter, centroid, and orthocenter are the same point. Construct altitudes through each vertex and label the intersection. Rotate the triangle about point D. A 120 degree rotation will map the image to the original. Show the labels of the image.

Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

There are 3 lines of symmetry.

b. Construct an equilateral triangle and show the labels of the vertices. Next, find the center of the triangle. Since this is an equilateral triangle, the circumcenter, incenter, centroid, and orthocenter are the same point. Construct altitudes through each Since the figure maps onto itself 3 times as it is vertex and label the intersection. rotated, the order of symmetry is 3. Rotate the triangle about point D. A 120 degree c. rotation will map the image to the original. Show the Square labels of the image. Construct a square and then construct lines through the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The The triangle can be rotated a third time about D. A image maps to the original at 72, 144, 216, 288, and 360 degree rotation maps the image to the original. 360 degree rotations. So the order of symmetry is 5.

Regular Pentagon Construct a regular pentagon and then construct lines

through each vertex perpendicular to the sides. Use d. Sample answer: for each figure studied, the the reflection tool first to find that the image maps number of sides of the figure is the same as the lines onto the original when reflected in each of the 5 lines of symmetry and the order of symmetry. A regular constructed. So there are 5 lines of symmetry. polygon with n sides has n lines of symmetry and Next, rotate the square about the center point. The order of symmetry n. image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5. ANSWER: a. 3 b. 3 c.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure also has rotational symmetry.

d. Sample answer: for each figure studied, the Therefore, neither of them are correct. Figure A has number of sides of the figure is the same as the lines both line and rotational symmetry. of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and ANSWER: order of symmetry n. Neither; Figure A has both line and rotational symmetry. ANSWER: a. 3 42. CHALLENGE A quadrilateral in the coordinate b. 3 plane has exactly two lines of symmetry, y = x – 1 c. and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of symmetry. SOLUTION: d. Sample answer: A regular polygon with n sides Graph the figure and the lines of symmetry. has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

Pick points that are the same distance a from one line and the same distance b from the other line. In the same answer, the quadrilateral is a rectangle with SOLUTION: sides which are parallel to the lines of symmetry. A figure has line symmetry if the figure can be This guarantees that the vertices of the quadrilateral mapped onto itself by a reflection in a line. This are the same distance a from one line and the same figure has 4 lines of symmetry. distance b from the other line. In this case, a =

and b = .

A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2).

The figure also has rotational symmetry.

Therefore, neither of them are correct. Figure A has 43. REASONING A figure has infinitely many lines of both line and rotational symmetry. symmetry. What is the figure? Explain. SOLUTION: ANSWER: circle; Every line through the center of a circle is a Neither; Figure A has both line and rotational line of symmetry, and there are infinitely many such symmetry. lines.

42. CHALLENGE A quadrilateral in the coordinate ANSWER: plane has exactly two lines of symmetry, y = x – 1 circle; Every line through the center of a circle is a and y = –x + 2. Find a set of possible vertices for line of symmetry, and there are infinitely many such the figure. Graph the figure and the lines of lines. symmetry. 44. OPEN-ENDED Draw a figure with line symmetry SOLUTION: but not rotational symmetry. Explain. Graph the figure and the lines of symmetry. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not

have rotational symmetry. Pick points that are the same distance a from one line and the same distance b from the other line. In An isosceles triangle has line symmetry from the the same answer, the quadrilateral is a rectangle with vertex angle to the base of the triangle, but it does sides which are parallel to the lines of symmetry. not have rotational symmetry because it cannot be This guarantees that the vertices of the quadrilateral rotated from 0° to 360° and map onto itself. are the same distance a from one line and the same distance b from the other line. In this case, a =

and b = .

A set of possible vertices for the figure are, (–1, 0), ANSWER: (2, 3), (4, 1), and (1, 2). Sample answer: An isosceles triangle has line symmetry from the vertex angle to the base of the ANSWER: triangle, but it does not have rotational symmetry Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) because it cannot be rotated from 0° to 360° and map onto itself.

45. WRITING IN MATH How are line symmetry and rotational symmetry related? 43. REASONING A figure has infinitely many lines of SOLUTION: symmetry. What is the figure? Explain. In both types of symmetries the figure is mapped onto itself. SOLUTION: circle; Every line through the center of a circle is a Rotational symmetry. line of symmetry, and there are infinitely many such lines.

ANSWER: Reflectional symmetry: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain.

Identify a figure that has line symmetry but does not have rotational symmetry.

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does ANSWER: not have rotational symmetry because it cannot be Sample answer: In both rotational and line symmetry rotated from 0° to 360° and map onto itself. a figure is mapped onto itself. However, in line symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry.

A 2 45. WRITING IN MATH How are line symmetry and B 3 rotational symmetry related? C 4 SOLUTION: D 8 In both types of symmetries the figure is mapped SOLUTION: onto itself.

Rotational symmetry.

Reflectional symmetry: The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn around its center.

2 + 2 = 4, so C is the correct answer. In some cases an object can have both rotational and reflectional symmetry, such as the diamond, however ANSWER: some objects do not have both such as the crab. C

47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A

B ANSWER: Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line

symmetry the figure is mapped onto itself by a C reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry. D 46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of symmetry and the order of symmetry, and then she E enters this value into a database. Which value should she enter in the database for the tile shown here?

SOLUTION: Option A has rotational and reflectional symmetry.

A 2 B 3 Option B has reflectional symmetry but not rotational symmetry. C 4 D 8 SOLUTION:

Option C has neither rotational nor reflectional symmetry.

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has Option D has rotational symmetry but not reflectional rotational symmetry at 180 degrees, or each half turn symmetry. around its center.

2 + 2 = 4, so C is the correct answer.

ANSWER:

C Option E has reflectional symmetry but not rotational symmetry. 47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A

B The correct choice is D.

ANSWER:

C D 48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? D A Equilateral triangle B Equiangular triangle C Isosceles triangle E D Scalene triangle SOLUTION: An isosceles triangle has one line of symmetry and SOLUTION: no rotational symmetry. The correct choice is C. Option A has rotational and reflectional symmetry. ANSWER: C

A B C D Option C has neither rotational nor reflectional symmetry. SOLUTION: First, plot the points.

Option D has rotational symmetry but not reflectional symmetry.

Option E has reflectional symmetry but not rotational symmetry.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

The correct choice is D.

ANSWER: D

48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle D Scalene triangle SOLUTION: An isosceles triangle has one line of symmetry and Option B has reflective symmetry but not rotational no rotational symmetry. The correct choice is C. symmetry. The correct choice is B.

ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D SOLUTION: First, plot the points.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

State whether the figure appears to have line Two lines of reflection go through the vertices of the symmetry. Write yes or no. If so, copy the figure. figure, draw all lines of symmetry, and state their number.

SOLUTION: Thus, there are four possible lines that go through A figure has reflectional symmetry if the figure can the center and are lines of reflections.

be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure. Therefore, the figure has four lines of symmetry.

ANSWER: yes; 4

Two lines of reflection go through the vertices of the figure.

SOLUTION: Thus, there are four possible lines that go through A figure has reflectional symmetry if the figure can the center and are lines of reflections. be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER: no

Therefore, the figure has four lines of symmetry.

ANSWER: yes; 4 3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

The figure has a vertical line of symmetry.

2. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. It does not have a horizontal line of symmetry.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER: The figure does not have a line of symmetry through no the vertices.

3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. Thus, the figure has only one line of symmetry.

ANSWER: The given figure has reflectional symmetry. yes; 1

The figure has a vertical line of symmetry.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the It does not have a horizontal line of symmetry. figure, locate the center of symmetry, and state the order and magnitude of symmetry.

The figure does not have a line of symmetry through 4. the vertices. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational Thus, the figure has only one line of symmetry. symmetry.

ANSWER: ANSWER: yes; 1 no

SOLUTION: State whether the figure has rotational A figure in the plane has rotational symmetry if the symmetry. Write yes or no. If so, copy the figure can be mapped onto itself by a rotation figure, locate the center of symmetry, and state between 0° and 360° about the center of the figure. the order and magnitude of symmetry.

The given figure has rotational symmetry.

4. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° The number of times a figure maps onto itself as it and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational rotates form 0° and 360° is called the order of symmetry. symmetry.

ANSWER: The given figure has order of symmetry of 2, since no the figure can be rotated twice in 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

5. Since the figure has order 2 rotational symmetry, the SOLUTION: magnitude of the symmetry is . A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation yes; 2; 180° between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

SOLUTION: The number of times a figure maps onto itself as it A figure in the plane has rotational symmetry if the rotates form is called the order of 0° and 360° figure can be mapped onto itself by a rotation symmetry. between 0° and 360° about the center of the figure.

The given figure has order of symmetry of 2, since The given figure has rotational symmetry. the figure can be rotated twice in 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is . The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of ANSWER: symmetry. yes; 2; 180° Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

6. ANSWER: SOLUTION: yes; 4; 90° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry 7.

SOLUTION: The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps Vertical and horizontal lines through the center and onto itself. diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way.

The figure has magnitude of symmetry of The equations of those lines in this figure are x = 0, . y = -1, y = x - 1, and y = -x - 1. ANSWER: Each quarter turn also maps the square onto itself. yes; 4; 90° So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the State whether the figure has line symmetry rotations of 90, 180, and 270 degrees around the point and/or rotational symmetry. If so, describe the (0, -1) map the square onto itself. reflections and/or rotations that map the figure onto itself.

8. 7. SOLUTION: SOLUTION: This figure does not have line symmetry, because Vertical and horizontal lines through the center and adjacent sides are not congruent. diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way. It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around The equations of those lines in this figure are x = 0, the point (1, 1) maps the parallelogram onto itself. y = -1, y = x - 1, and y = -x - 1. ANSWER: Each quarter turn also maps the square onto itself. rotational symmetry; the rotation of 180 degrees So the rotations of 90, 180, and 270 degrees around around the point (1, 1) maps the parallelogram onto the point (0, -1) map the square onto itself. itself.

ANSWER: REGULARITY State whether the figure line symmetry; rotational symmetry; the reflection in appears to have line symmetry. Write yes or no. the line x = 0, the reflection in the line y = -1, the If so, copy the figure, draw all lines of reflection in the line y = x - 1, and the reflection in symmetry, and state their number. the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

9. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection 8. where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry. SOLUTION: This figure does not have line symmetry, because ANSWER: adjacent sides are not congruent. no

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

ANSWER: 10. rotational symmetry; the rotation of 180 degrees around the point (1, 1) maps the parallelogram onto SOLUTION: itself. A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line. REGULARITY State whether the figure appears to have line symmetry. Write yes or no. The given figure has reflectional symmetry. If so, copy the figure, draw all lines of symmetry, and state their number. In order for the figure to map onto itself, the line of reflection must go through the center point.

The figure has a vertical and horizontal line of 9. reflection. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure

does not have any lines of of symmetry. It is also possible to have reflection over the ANSWER: diagonal lines. no

10. Therefore, the figure has four lines of symmetry SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point. ANSWER: The figure has a vertical and horizontal line of yes; 4 reflection.

It is also possible to have reflection over the diagonal lines.

11.

SOLUTION: A figure has reflectional symmetry if the figure can Therefore, the figure has four lines of symmetry be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though opposites edges. ANSWER: yes; 4

There are three lines of reflection that go though opposites vertices.

11. There are six possible lines that go through the center SOLUTION: and are lines of reflections. Thus, the hexagon has A figure has reflectional symmetry if the figure can six lines of symmetry. be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though opposites edges. ANSWER: yes; 6

There are three lines of reflection that go though opposites vertices.

12. SOLUTION: A figure has reflectional symmetry if the figure can There are six possible lines that go through the center be mapped onto itself by a reflection in a line. and are lines of reflections. Thus, the hexagon has six lines of symmetry. The figure has reflectional symmetry.

There is only one line of symmetry, a horizontal line through the middle of the figure.

Thus, the figure has one line of symmetry. ANSWER: yes; 6 ANSWER: yes; 1

13. SOLUTION: 12. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. SOLUTION: A figure has reflectional symmetry if the figure can The figure has reflectional symmetry. be mapped onto itself by a reflection in a line. There is only one possible line of reflection, The figure has reflectional symmetry. horizontally though the middle of the figure.

There is only one line of symmetry, a horizontal line through the middle of the figure.

Thus, the figure has one line of symmetry.

ANSWER:

yes; 1 Thus, the figure has one line of symmetry.

ANSWER: yes; 1

14. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. 13. SOLUTION: The given figure does not have reflectional symmetry. It is not possible to draw a line of A figure has reflectional symmetry if the figure can reflection where the figure can map onto itself. be mapped onto itself by a reflection in a line. ANSWER: The figure has reflectional symmetry. no

There is only one possible line of reflection, FLAGS State whether each flag design appears to horizontally though the middle of the figure. have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their number. 15. Refer to page 262.

Thus, the figure has one line of symmetry. SOLUTION: A figure has reflectional symmetry if the figure can ANSWER: be mapped onto itself by a reflection in a line. yes; 1 The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER: no 14. SOLUTION: 16. Refer to the flag on page 262. A figure has reflectional symmetry if the figure can SOLUTION: be mapped onto itself by a reflection in a line. A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line. The given figure does not have reflectional symmetry. It is not possible to draw a line of The figure has reflectional symmetry. reflection where the figure can map onto itself.

ANSWER: In order for the figure to map onto itself, the line of no reflection must go through the center point.

FLAGS State whether each flag design appears to A horizontal and vertical lines of reflection are have line symmetry. Write yes or no. If so, copy the possible. flag, draw all lines of symmetry, and state their number. 15. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same

location above and below the center horizontal line, Two diagonal lines of reflection are possible. the flag would have three lines of symmetry.

ANSWER: no

16. Refer to the flag on page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. There are a total of four possible lines that go through the center and are lines of reflections. Thus, In order for the figure to map onto itself, the line of the flag has four lines of symmetry. reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

ANSWER: yes; 4

Two diagonal lines of reflection are possible.

17. Refer to page 262.

There are a total of four possible lines that go SOLUTION: through the center and are lines of reflections. Thus, A figure has reflectional symmetry if the figure can the flag has four lines of symmetry. be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

It is not possible to reflect over a vertical or line ANSWER: through the diagonals. yes; 4 Thus, the figure has one line of symmetry.

ANSWER: yes; 1

17. Refer to page 262. State whether the figure has rotational symmetry. Write yes or no. If so, copy the SOLUTION: figure, locate the center of symmetry, and state A figure has reflectional symmetry if the figure can the order and magnitude of symmetry. be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. 18. A horizontal line is a line of reflections for this flag. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry. The figure has rotational symmetry. ANSWER: yes; 1 The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto itself.

18. ANSWER: SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

yes; 2; 180°

The figure has rotational symmetry. 19. SOLUTION: The number of times a figure maps onto itself as it A figure in the plane has rotational symmetry if the rotates from 0° to 360° is called the order of figure can be mapped onto itself by a rotation symmetry. between 0° and 360° about the center of the figure.

This figure has order 2 rotational symmetry, since The triangle has rotational symmetry. you have to rotate 180° to get the figure to map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has a magnitude of symmetry of . The number of times a figure maps onto itself as it ANSWER: rotates from 0° to 360° is called the order of symmetry.

The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle yes; 2; 180° through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

19. ANSWER: SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The triangle has rotational symmetry. yes; 3; 120°

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The number of times a figure maps onto itself as it between 0° and 360° about the center of the figure. rotates from 0° to 360° is called the order of symmetry. The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be The figure has order 3 rotational symmetry. mapped onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps ANSWER: onto itself. no

The figure has magnitude of symmetry of .

ANSWER: 21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. yes; 3; 120° The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

ANSWER: no

21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation

between 0° and 360° about the center of the figure. ANSWER: The crescent shaped figure has no rotational no symmetry. There is no way to rotate it such that it can be mapped onto itself.

22. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it ANSWER: map onto itself within 360°. no The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

22. The figure has magnitude of symmetry of . SOLUTION: A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

yes; 8; 45°

23. The number of times a figure maps onto itself as it SOLUTION: rotates from 0° to 360° is called the order of A figure in the plane has rotational symmetry if the

symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it The figure has rotational symmetry. map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of . The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of ANSWER: symmetry.

The figure has order 8 rotational symmetry. This means that the figure can be rotated 8 times and map onto itself within 360°.

The magnitude of symmetry is the smallest angle yes; 8; 45° through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

23. ANSWER: SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

yes; 8; 45° The figure has rotational symmetry. WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, state the order and magnitude of symmetry. 24. Refer to page 263. SOLUTION: A figure in the plane has rotational symmetry if the The number of times a figure maps onto itself as it figure can be mapped onto itself by a rotation rotates from 0° to 360° is called the order of between 0° and 360° about the center of the figure. symmetry. The wheel has rotational symmetry. The figure has order 8 rotational symmetry. This means that the figure can be rotated 8 times and map The number of times a figure maps onto itself as it

onto itself within 360°. rotates from 0° to 360° is called the order of symmetry. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The wheel has order 5 rotational symmetry. There

onto itself. are 5 large spokes and 5 small spokes. You can rotate the wheel 5 times within 360° and map the The figure has magnitude of symmetry of figure onto itself. . The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps onto itself.

The wheel has magnitude of symmetry . yes; 8; 45° ANSWER: WHEELS State whether each wheel cover appears yes; 5; 72° to have rotational symmetry. Write yes or no. If so, state the order and magnitude of symmetry. 25. Refer to page 263. 24. Refer to page 263. SOLUTION: SOLUTION: A figure in the plane has rotational symmetry if the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. between 0° and 360° about the center of the figure. The wheel has rotational symmetry. The wheel has rotational symmetry. The number of times a figure maps onto itself as it The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of rotates from 0° to 360° is called the order of symmetry. symmetry. The wheel has order 8 rotational symmetry. There The wheel has order 5 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times are 5 large spokes and 5 small spokes. You can within 360° and map onto itself. rotate the wheel 5 times within 360° and map the figure onto itself. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The magnitude of symmetry is the smallest angle onto itself. through which a figure can be rotated so that it maps onto itself. The wheel has order 8 rotational symmetry and

magnitude . The wheel has magnitude of symmetry . ANSWER: yes; 8; 45° ANSWER: yes; 5; 72° 26. Refer to page 263. SOLUTION: 25. Refer to page 263. A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation A figure in the plane has rotational symmetry if the between 0° and 360° about the center of the figure. figure can be mapped onto itself by a rotation The wheel has rotational symmetry. between 0° and 360° about the center of the figure. The number of times a figure maps onto itself as it The wheel has rotational symmetry. rotates from 0° to 360° is called the order of symmetry. The wheel has order 10 rotational The number of times a figure maps onto itself as it symmetry. There are 10 bolts and the tire can be rotates from 0° to 360° is called the order of rotated 10 times within 360° and map onto itself. symmetry. The magnitude of symmetry is the smallest angle The wheel has order 8 rotational symmetry. There through which a figure can be rotated so that it maps are 8 spokes, thus the wheel can be rotated 8 times onto itself. The wheel has magnitude of symmetry of within 360° and map onto itself. .

ANSWER: The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps yes; 10; 36°

onto itself. State whether the figure has line symmetry and/or rotational symmetry. If so, describe the The wheel has order 8 rotational symmetry and reflections and/or rotations that map the figure magnitude . onto itself.

ANSWER: yes; 8; 45° 26. Refer to page 263. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 27. between 0° and 360° about the center of the figure. SOLUTION: The wheel has rotational symmetry. This triangle is scalene, so it cannot have symmetry. The number of times a figure maps onto itself as it ANSWER: rotates from 0° to 360° is called the order of no symmetry symmetry. The wheel has order 10 rotational symmetry. There are 10 bolts and the tire can be rotated 10 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The wheel has magnitude of symmetry of . 28. ANSWER: SOLUTION: yes; 10; 36° This figure is a square, because each pair of adjacent State whether the figure has line symmetry sides is congruent and perpendicular. and/or rotational symmetry. If so, describe the All squares have both line and rotational symmetry. reflections and/or rotations that map the figure The line symmetry is vertically, horizontally, and onto itself. diagonally through the center of the square, with lines that are either parallel to the sides of the square or that include two vertices of the square. The equations of those lines are: x = 0, y = 0, y = x, and y = -x

The rotational symmetry is for each quarter turn in a square, so the rotations of 90, 180, and 270 degrees around the origin map the square onto itself.

27. SOLUTION: ANSWER: This triangle is scalene, so it cannot have symmetry. line symmetry; rotational symmetry; the reflection in ANSWER: the line x = 0, the reflection in the line y = 0, the no symmetry reflection in the line y = x, and the reflection in the line y = -x all map the square onto itself; the rotations of 90, 180, and 270 degrees around the origin map the square onto itself.

28. SOLUTION: 29. This figure is a square, because each pair of adjacent sides is congruent and perpendicular. SOLUTION: All squares have both line and rotational symmetry. The trapezoid has line symmetry, because it is The line symmetry is vertically, horizontally, and isosceles, but it does not have rotational symmetry, diagonally through the center of the square, with lines because no trapezoid does. that are either parallel to the sides of the square or that include two vertices of the square. The The reflection in the line y = 1.5 maps the trapezoid equations of those lines are: x = 0, y = 0, y = x, and y onto itself, because that is the perpendicular bisector = -x to the parallel sides.

The rotational symmetry is for each quarter turn in a ANSWER: square, so the rotations of 90, 180, and 270 degrees line symmetry; the reflection in the line y = 1.5 maps around the origin map the square onto itself. the trapezoid onto itself.

ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = 0, the reflection in the line y = x, and the reflection in the line y = -x all map the square onto itself; the rotations of 90, 180, and 270 degrees around the origin map the square onto itself. 30. SOLUTION: This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its center, which is the point (1, -1.5).

Since this parallelogram is not a rhombus it does not have line symmetry. 29. SOLUTION: ANSWER: The trapezoid has line symmetry, because it is rotational symmetry; the rotation of 180 degrees isosceles, but it does not have rotational symmetry, around the point (1, -1.5) maps the parallelogram

because no trapezoid does. onto itself.

31. MODELING Symmetry is an important component The reflection in the line y = 1.5 maps the trapezoid of photography. Photographers often use reflection in onto itself, because that is the perpendicular bisector water to create symmetry in photos. The photo on to the parallel sides. page 263 is a long exposure shot of the Eiffel tower ANSWER: reflected in a pool. line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself. a. Describe the two-dimensional symmetry created by the photo. b. Is there rotational symmetry in the photo? Explain your reasoning. SOLUTION: a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the

30. center of the photo. SOLUTION: b No; sample answer: Because of how the image is This figure is a parallelogram, so it has rotational reflected over the horizontal line, there is no symmetry of a half turn or 180 degrees around its rotational symmetry. center, which is the point (1, -1.5). ANSWER:

Since this parallelogram is not a rhombus it does not a. Sample answer: There is a horizontal line of have line symmetry. symmetry between the tower and its reflection. There is a vertical line of symmetry through the ANSWER: center of the photo. rotational symmetry; the rotation of 180 degrees b No; sample answer: Because of how the image is 3-5 Symmetryaround the point (1, -1.5) maps the parallelogram reflected over the horizontal line, there is no onto itself. rotational symmetry.

31. MODELING Symmetry is an important component COORDINATE GEOMETRY Determine of photography. Photographers often use reflection in whether the figure with the given vertices has line water to create symmetry in photos. The photo on symmetry and/or rotational symmetry. page 263 is a long exposure shot of the Eiffel tower 32. R(–3, 3), S(–3, –3), T(3, 3) reflected in a pool. SOLUTION:

Draw the figure on a coordinate plane. a. Describe the two-dimensional symmetry created by the photo. b. Is there rotational symmetry in the photo? Explain your reasoning. SOLUTION: a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is A figure has line symmetry if the figure can be reflected over the horizontal line, there is no mapped onto itself by a reflection in a line. rotational symmetry. The given triangle has a line of symmetry through ANSWER: points (0, 0) and (–3, 3). a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. A figure in the plane has rotational symmetry if the There is a vertical line of symmetry through the figure can be mapped onto itself by a rotation center of the photo. between 0° and 360° about the center of the figure. b No; sample answer: Because of how the image is There is not way to rotate the figure and have it map reflected over the horizontal line, there is no onto itself. rotational symmetry. Thus, the figure has only line symmetry. COORDINATE GEOMETRY Determine whether the figure with the given vertices has line ANSWER: symmetry and/or rotational symmetry. line 32. R(–3, 3), S(–3, –3), T(3, 3) SOLUTION: 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) Draw the figure on a coordinate plane. SOLUTION: Draw the figure on a coordinate plane.

A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure has line symmetry if the figure can be The given triangle has a line of symmetry through mapped onto itself by a reflection in a line. The points (0, 0) and (–3, 3). given figure has 4 lines of symmetry. The line of symmetry are though the following pairs of points

{(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, A figure in the plane has rotational symmetry if the eSolutions Manual - Powered by Cognero and {(2, 2), (2, –2)}. Page 13 figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. A figure in the plane has rotational symmetry if the There is not way to rotate the figure and have it map figure can be mapped onto itself by a rotation onto itself. between 0° and 360° about the center of the figure.

The figure can be rotated from the origin and map Thus, the figure has only line symmetry. onto itself. The order of symmetry is 4. ANSWER: Thus, the figure has both symmetry and line line rotational symmetry.

33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) ANSWER: SOLUTION: line and rotational Draw the figure on a coordinate plane. 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 2) SOLUTION: Draw the figure on a coordinate plane.

A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The given figure has 4 lines of symmetry. The line of

symmetry are though the following pairs of points A figure has line symmetry if the figure can be {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, mapped onto itself by a reflection in a line. The

and {(2, 2), (2, –2)}. given hexagon has 2 lines of symmetry. The lines pass through the following pair of points {(0, 4), (0, – A figure in the plane has rotational symmetry if the 4)}, and {(3, 0), (–3, 0)} figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. A figure in the plane has rotational symmetry if the The figure can be rotated from the origin and map figure can be mapped onto itself by a rotation

onto itself. The order of symmetry is 4. between 0° and 360° about the center of the figure. The figure has rotational symmetry. You can rotate Thus, the figure has both line symmetry and the figure once within 360° and have it map to itself. rotational symmetry. Thus, the figure has both line symmetry and ANSWER: rotational symmetry. line and rotational ANSWER: 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 2) line and rotational SOLUTION: 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) Draw the figure on a coordinate plane. SOLUTION: Draw the figure on a coordinate plane.

A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The given hexagon has 2 lines of symmetry. The lines A figure has line symmetry if the figure can be pass through the following pair of points {(0, 4), (0, – mapped onto itself by a reflection in a line. The 4)}, and {(3, 0), (–3, 0)} trapezoid has a line of reflection through points (0,3) and (0, –3). A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation A figure in the plane has rotational symmetry if the between 0° and 360° about the center of the figure. figure can be mapped onto itself by a rotation The figure has rotational symmetry. You can rotate between 0° and 360° about the center of the the figure once within 360° and have it map to itself. figure. There is no way to rotate this figure and have it map onto itself. Thus, it does not have rotational Thus, the figure has both line symmetry and symmetry. rotational symmetry. Therefore, the figure has only line symmetry. ANSWER: line and rotational ANSWER: line 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) ALGEBRA Graph the function and determine SOLUTION: whether the graph has line and/or rotational Draw the figure on a coordinate plane. symmetry. If so, state the order and magnitude of symmetry, and write the equations of any lines of symmetry. 36. y = x SOLUTION: Graph the function.

A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The trapezoid has a line of reflection through points (0,3) and (0, –3).

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the A figure has reflectional symmetry if the figure can figure. There is no way to rotate this figure and have be mapped onto itself by a reflection in a line. The it map onto itself. Thus, it does not have rotational line y = x has reflectional symmetry since any line symmetry. perpendicular to y = x is a line of reflection. The equation of the line symmetry is y = –x. Therefore, the figure has only line symmetry. ANSWER: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation line between 0° and 360° about the center of the figure. The line can be rotated twice within 360° and be ALGEBRA Graph the function and determine mapped onto itself. whether the graph has line and/or rotational symmetry. If so, state the order and magnitude of symmetry, and write the equations of any lines of The number of times a figure maps onto itself as it symmetry. rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational 36. y = x symmetry. SOLUTION: Graph the function. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of .

Thus, the graph has both reflectional and rotational symmetry.

ANSWER: rotational; 2; 180°; line symmetry; y = –x A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The line y = x has reflectional symmetry since any line perpendicular to y = x is a line of reflection. The equation of the line symmetry is y = –x.

The number of times a figure maps onto itself as it SOLUTION: rotates from 0° to 360° is called the order of Graph the function. symmetry. The graph has order 2 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of .

Thus, the graph has both reflectional and rotational symmetry. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The ANSWER: graph is reflected through the y-axis. Thus, the rotational; 2; 180°; line symmetry; y = –x equation of the line symmetry is x = 0.

Thus, the graph has only reflectional symmetry.

ANSWER:

2 line; x = 0 37. y = x + 1 SOLUTION: Graph the function.

38. y = –x3 SOLUTION: Graph the function. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph is reflected through the y-axis. Thus, the equation of the line symmetry is x = 0.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. There is no way to rotate the graph and have it map onto itself. A figure has reflectional symmetry if the figure can Thus, the graph has only reflectional symmetry. be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the ANSWER: graph can be mapped onto itself. line; x = 0 A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. You can rotate the graph through the origin and have it map onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry. 3 38. y = –x SOLUTION: The magnitude of symmetry is the smallest angle Graph the function. through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of .

Thus, the graph has only rotational symmetry.

ANSWER: rotational; 2; 180°

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the 39. Refer to the rectangle on the coordinate plane. figure. You can rotate the graph through the origin and have it map onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry.

a. The magnitude of symmetry is the smallest angle What are the equations of the lines of symmetry of

through which a figure can be rotated so that it maps the rectangle? b. What happens to the equations of the lines of onto itself. The graph has magnitude of symmetry of . symmetry when the rectangle is rotated 90 degrees counterclockwise around its center of symmetry? Thus, the graph has only rotational symmetry. Explain. SOLUTION: ANSWER: a. The lines of symmetry are parallel to the sides of rotational; 2; 180° the rectangles, and through the center of rotation.

The slopes of the sides of the rectangle are 0.5 and -2, so the slopes of the lines of symmetry are the same. The center of the rectangle is (1, 1.5). Use the point-slope formula to find equations.

39. Refer to the rectangle on the coordinate plane. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

ANSWER: a. b. The equations of the lines of symmetry do not a. What are the equations of the lines of symmetry of change; although the rectangle does not map onto the rectangle? itself under this rotation, the lines of symmetry are b. What happens to the equations of the lines of mapped to each other. symmetry when the rectangle is rotated 90 degrees counterclockwise around its center of symmetry? 40. MULTIPLE REPRESENTATIONS In this Explain. problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular SOLUTION: polygons. a. The lines of symmetry are parallel to the sides of the rectangles, and through the center of rotation. a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool The slopes of the sides of the rectangle are 0.5 and under the transformation menu to investigate and -2, so the slopes of the lines of symmetry are the determine all possible lines of symmetry. Then record same. their number. b. Geometric Use the rotation tool under the The center of the rectangle is (1, 1.5). Use the transformation menu to investigate the rotational point-slope formula to find equations. symmetry of the figure in part a. Then record its

order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. b. The equations of the lines of symmetry do not Record the number of lines of symmetry and the order of symmetry for each polygon. change; although the rectangle does not map onto d. Verbal Make a conjecture about the number of itself under this rotation, the lines of symmetry are lines of symmetry and the order of symmetry for a mapped to each other. regular polygon with n sides. ANSWER: SOLUTION: a. a. Construct an equilateral triangle and label the b. The equations of the lines of symmetry do not vertices A, B, and C. Draw a line through A change; although the rectangle does not map onto perpendicular to . Reflect the triangle in the line. itself under this rotation, the lines of symmetry are Show the labels of the reflected image. If the image mapped to each other. maps to the original, then this line is a line of reflection. 40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational Next, draw a line through B perpendicular to . symmetry of the figure in part a. Then record its Reflect the triangle in the line. Show the labels of the order of symmetry. reflected image. If the image maps to the original, c. Tabular Repeat the process in parts a and b for a then this line is a line of reflection. square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides. SOLUTION: a. Construct an equilateral triangle and label the

vertices A, B, and C. Draw a line through A Lastly, draw a line through C perpendicular to . perpendicular to . Reflect the triangle in the line. Reflect the triangle in the line. Show the labels of the Show the labels of the reflected image. If the image reflected image. If the image maps to the original, maps to the original, then this line is a line of then this line is a line of reflection. reflection.

Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

There are 3 lines of symmetry.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

There are 3 lines of symmetry.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

Regular Pentagon Construct a regular pentagon and then construct lines Since the figure maps onto itself 3 times as it is through each vertex perpendicular to the sides. Use rotated, the order of symmetry is 3. the reflection tool first to find that the image maps c. onto the original when reflected in each of the 5 lines Square constructed. So there are 5 lines of symmetry. Construct a square and then construct lines through Next, rotate the square about the center point. The the midpoints of each side and diagonals. Use the image maps to the original at 72, 144, 216, 288, and reflection tool first to find that the image maps onto 360 degree rotations. So the order of symmetry is 5. the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Regular Hexagon Construct a regular hexagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps Regular Pentagon onto the original when reflected in each of the 6 lines Construct a regular pentagon and then construct lines constructed. So there are 6 lines of symmetry. through each vertex perpendicular to the sides. Use Next, rotate the square about the center point. The the reflection tool first to find that the image maps image maps to the original at 60, 120, 180, 240, 300, onto the original when reflected in each of the 5 lines and 360 degree rotations. So the order of symmetry constructed. So there are 5 lines of symmetry. is 6. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5.

d. Sample answer: for each figure studied, the

number of sides of the figure is the same as the lines Regular Hexagon of symmetry and the order of symmetry. A regular Construct a regular hexagon and then construct lines polygon with n sides has n lines of symmetry and through each vertex perpendicular to the sides. Use order of symmetry n. the reflection tool first to find that the image maps onto the original when reflected in each of the 6 lines ANSWER: constructed. So there are 6 lines of symmetry. a. 3 Next, rotate the square about the center point. The b. 3 image maps to the original at 60, 120, 180, 240, 300, c. and 360 degree rotations. So the order of symmetry is 6.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

SOLUTION: A figure has line symmetry if the figure can be d. Sample answer: for each figure studied, the mapped onto itself by a reflection in a line. This number of sides of the figure is the same as the lines figure has 4 lines of symmetry. of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and order of symmetry n.

ANSWER: a. 3 b. 3 c. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them The figure also has rotational symmetry. correct? Explain your reasoning.

Therefore, neither of them are correct. Figure A has both line and rotational symmetry.

ANSWER: Neither; Figure A has both line and rotational symmetry. SOLUTION: A figure has line symmetry if the figure can be 42. CHALLENGE A quadrilateral in the coordinate mapped onto itself by a reflection in a line. This plane has exactly two lines of symmetry, y = x – 1 figure has 4 lines of symmetry. and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of symmetry. SOLUTION: Graph the figure and the lines of symmetry.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Pick points that are the same distance a from one line and the same distance b from the other line. In the same answer, the quadrilateral is a rectangle with The figure also has rotational symmetry. sides which are parallel to the lines of symmetry. This guarantees that the vertices of the quadrilateral Therefore, neither of them are correct. Figure A has are the same distance a from one line and the same both line and rotational symmetry. distance b from the other line. In this case, a = ANSWER: Neither; Figure A has both line and rotational and b = . symmetry. A set of possible vertices for the figure are, (–1, 0), 42. CHALLENGE A quadrilateral in the coordinate (2, 3), (4, 1), and (1, 2). plane has exactly two lines of symmetry, y = x – 1 ANSWER: and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) symmetry. SOLUTION: Graph the figure and the lines of symmetry.

43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. SOLUTION: Pick points that are the same distance a from one circle; Every line through the center of a circle is a line and the same distance b from the other line. In line of symmetry, and there are infinitely many such the same answer, the quadrilateral is a rectangle with lines. sides which are parallel to the lines of symmetry. This guarantees that the vertices of the quadrilateral ANSWER: are the same distance a from one line and the same circle; Every line through the center of a circle is a distance b from the other line. In this case, a = line of symmetry, and there are infinitely many such lines. and b = . 44. OPEN-ENDED Draw a figure with line symmetry

but not rotational symmetry. Explain. A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2). SOLUTION: A figure has line symmetry if the figure can be ANSWER: mapped onto itself by a reflection in a line. A figure Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not have rotational symmetry.

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself. 43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. SOLUTION: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines. ANSWER: ANSWER: Sample answer: An isosceles triangle has line circle; Every line through the center of a circle is a symmetry from the vertex angle to the base of the line of symmetry, and there are infinitely many such triangle, but it does not have rotational symmetry lines. because it cannot be rotated from 0° to 360° and map onto itself. 44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can 45. WRITING IN MATH How are line symmetry and be mapped onto itself by a rotation between 0° and rotational symmetry related? 360° about the center of the figure. SOLUTION: Identify a figure that has line symmetry but does not In both types of symmetries the figure is mapped have rotational symmetry. onto itself.

An isosceles triangle has line symmetry from the Rotational symmetry. vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself. Reflectional symmetry:

ANSWER: Sample answer: An isosceles triangle has line In some cases an object can have both rotational and symmetry from the vertex angle to the base of the reflectional symmetry, such as the diamond, however triangle, but it does not have rotational symmetry some objects do not have both such as the crab. because it cannot be rotated from 0° to 360° and map onto itself.

45. WRITING IN MATH How are line symmetry and ANSWER: rotational symmetry related? Sample answer: In both rotational and line symmetry SOLUTION: a figure is mapped onto itself. However, in line symmetry the figure is mapped onto itself by a In both types of symmetries the figure is mapped reflection, and in rotational symmetry, a figure is onto itself. mapped onto itself by a rotation. A figure can have

line symmetry and rotational symmetry. Rotational symmetry. 46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of Reflectional symmetry: symmetry and the order of symmetry, and then she enters this value into a database. Which value should she enter in the database for the tile shown here?

In some cases an object can have both rotational and reflectional symmetry, such as the diamond, however

some objects do not have both such as the crab. A 2

B 3 C 4 D 8 SOLUTION:

ANSWER: Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line The tile is a rhombus and has 2 lines of symmetry. symmetry the figure is mapped onto itself by a Each connects opposite corners of the tile. reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have It has an order of symmetry of 2, because it has line symmetry and rotational symmetry. rotational symmetry at 180 degrees, or each half turn around its center. 46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of 2 + 2 = 4, so C is the correct answer. symmetry and the order of symmetry, and then she enters this value into a database. Which value should ANSWER: she enter in the database for the tile shown here? C

47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A

A 2 B B 3 C 4 D 8

SOLUTION: C

The tile is a rhombus and has 2 lines of symmetry. E Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn SOLUTION: around its center. Option A has rotational and reflectional symmetry.

2 + 2 = 4, so C is the correct answer.

ANSWER: C Option B has reflectional symmetry but not rotational 47. Patrick drew a figure that has rotational symmetry symmetry. but not line symmetry. Which of the following could be the figure that Patrick drew? A

B Option C has neither rotational nor reflectional symmetry.

Option D has rotational symmetry but not reflectional symmetry. D

Option E has reflectional symmetry but not rotational symmetry. SOLUTION: Option A has rotational and reflectional symmetry.

Option B has reflectional symmetry but not rotational symmetry. The correct choice is D. ANSWER: D

48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry?

Option C has neither rotational nor reflectional A Equilateral triangle symmetry. B Equiangular triangle C Isosceles triangle D Scalene triangle

SOLUTION: Option D has rotational symmetry but not reflectional An isosceles triangle has one line of symmetry and symmetry. no rotational symmetry. The correct choice is C.

ANSWER: C

Option E has reflectional symmetry but not rotational 49. Camryn plotted the points , symmetry. and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D

The correct choice is D. SOLUTION: First, plot the points. ANSWER: D

Then, plot each option A-D to consider each figure ANSWER: and its symmetry. C Option A has both reflectional and rotational symmetry. 49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D SOLUTION: First, plot the points.

Option B has reflective symmetry but not rotational symmetry. The correct choice is B.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

The figure has reflectional symmetry.

Two lines of reflection go through the vertices of the figure.

ANSWER: no

Therefore, the figure has four lines of symmetry. 3.

ANSWER: SOLUTION: yes; 4 A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

The figure has a vertical line of symmetry.

2. It does not have a horizontal line of symmetry. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto The figure does not have a line of symmetry through itself. the vertices. ANSWER: no

3. Thus, the figure has only one line of symmetry. SOLUTION: ANSWER: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. yes; 1

The given figure has reflectional symmetry.

The figure has a vertical line of symmetry.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

It does not have a horizontal line of symmetry.

4. SOLUTION:

The figure does not have a line of symmetry through A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation the vertices. between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: Thus, the figure has only one line of symmetry. no

ANSWER: yes; 1

symmetry. Write yes or no. If so, copy the The given figure has rotational symmetry. figure, locate the center of symmetry, and state the order and magnitude of symmetry.

4. SOLUTION:

ANSWER: The magnitude of symmetry is the smallest angle no through which a figure can be rotated so that it maps onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is .

5. ANSWER: SOLUTION: yes; 2; 180° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

6. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation

The number of times a figure maps onto itself as it between 0° and 360° about the center of the figure.

rotates form 0° and 360° is called the order of The given figure has rotational symmetry. symmetry.

The given figure has order of symmetry of 2, since the figure can be rotated twice in 360°.

The magnitude of symmetry is the smallest angle

through which a figure can be rotated so that it maps onto itself. The number of times a figure maps onto itself as it

rotates from 0° to 360° is called the order of Since the figure has order 2 rotational symmetry, the symmetry. magnitude of the symmetry is . Since the figure can be rotated 4 times within 360° , ANSWER: it has order 4 rotational symmetry yes; 2; 180° The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER: yes; 4; 90° 6. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry. State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. 7.

SOLUTION: Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry Vertical and horizontal lines through the center and diagonal lines through two opposite vertices are all The magnitude of symmetry is the smallest angle lines of symmetry for a square oriented this way. through which a figure can be rotated so that it maps onto itself. The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1.

The figure has magnitude of symmetry of Each quarter turn also maps the square onto itself. . So the rotations of 90, 180, and 270 degrees around ANSWER: the point (0, -1) map the square onto itself. yes; 4; 90° ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

8. SOLUTION: This figure does not have line symmetry, because 7. adjacent sides are not congruent.

(0, -1) map the square onto itself. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry.

ANSWER: 8. no SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent.

REGULARITY State whether the figure In order for the figure to map onto itself, the line of appears to have line symmetry. Write yes or no. reflection must go through the center point. If so, copy the figure, draw all lines of

symmetry, and state their number. The figure has a vertical and horizontal line of reflection.

9. SOLUTION: A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line.

It is also possible to have reflection over the For the given figure, there are no lines of reflection diagonal lines. where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry.

ANSWER: no

Therefore, the figure has four lines of symmetry

10. SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry. ANSWER: In order for the figure to map onto itself, the line of yes; 4 reflection must go through the center point.

The figure has a vertical and horizontal line of reflection.

It is also possible to have reflection over the diagonal lines. 11. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given hexagon has reflectional symmetry. Therefore, the figure has four lines of symmetry In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though opposites edges.

ANSWER: yes; 4

There are three lines of reflection that go though opposites vertices.

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point. ANSWER: There are three lines of reflection that go though yes; 6 opposites edges.

There are three lines of reflection that go though opposites vertices. 12. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

Thus, the figure has one line of symmetry.

ANSWER:

yes; 1 ANSWER: yes; 6

13. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

12. The figure has reflectional symmetry.

The figure has reflectional symmetry.

There is only one line of symmetry, a horizontal line Thus, the figure has one line of symmetry. through the middle of the figure. ANSWER: yes; 1

Thus, the figure has one line of symmetry.

ANSWER: yes; 1 14. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. It is not possible to draw a line of 13. reflection where the figure can map onto itself.

ANSWER:

be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

The given figure does not have reflectional symmetry. It is not possible to draw a line of In order for the figure to map onto itself, the line of

reflection where the figure can map onto itself. reflection must go through the center point.

ANSWER: A horizontal and vertical lines of reflection are no possible.

ANSWER: no

16. Refer to the flag on page 262.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are possible.

ANSWER: yes; 4

Two diagonal lines of reflection are possible.

It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry. ANSWER: yes; 4 ANSWER: yes; 1

SOLUTION: The figure has reflectional symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation A horizontal line is a line of reflections for this flag. between 0° and 360° about the center of the figure.

It is not possible to reflect over a vertical or line through the diagonals. The figure has rotational symmetry.

18. SOLUTION: A figure in the plane has rotational symmetry if the

figure can be mapped onto itself by a rotation yes; 2; 180° between 0° and 360° about the center of the figure.

19. SOLUTION: A figure in the plane has rotational symmetry if the

The figure has rotational symmetry. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The triangle has rotational symmetry.

symmetry.

This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto itself.

ANSWER: The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

yes; 2; 180° The figure has magnitude of symmetry of .

ANSWER:

19. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation yes; 3; 120° between 0° and 360° about the center of the figure.

The triangle has rotational symmetry.

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has order 3 rotational symmetry. ANSWER: no The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The crescent shaped figure has no rotational symmetry. There is no way to rotate it such that it yes; 3; 120° can be mapped onto itself.

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

ANSWER: no

21. SOLUTION: A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation no between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps no onto itself.

The figure has magnitude of symmetry of .

ANSWER: 22. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry. yes; 8; 45°

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

yes; 8; 45° The figure has magnitude of symmetry of .

ANSWER:

23. SOLUTION:

The number of times a figure maps onto itself as it The wheel has rotational symmetry. rotates from 0° to 360° is called the order of symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The figure has order 8 rotational symmetry. This symmetry. means that the figure can be rotated 8 times and map onto itself within 360°. The wheel has order 5 rotational symmetry. There are 5 large spokes and 5 small spokes. You can The magnitude of symmetry is the smallest angle rotate the wheel 5 times within 360° and map the through which a figure can be rotated so that it maps figure onto itself. onto itself. The magnitude of symmetry is the smallest angle The figure has magnitude of symmetry of through which a figure can be rotated so that it maps . onto itself.

ANSWER: The wheel has magnitude of symmetry .

rotates from 0° to 360° is called the order of

The wheel has rotational symmetry. symmetry.

symmetry. within 360° and map onto itself.

figure onto itself.

The wheel has order 8 rotational symmetry and The magnitude of symmetry is the smallest angle magnitude . through which a figure can be rotated so that it maps onto itself. ANSWER:

yes; 8; 45° The wheel has magnitude of symmetry . 26. Refer to page 263. ANSWER: SOLUTION: yes; 5; 72° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. 25. Refer to page 263. The wheel has rotational symmetry. SOLUTION: A figure in the plane has rotational symmetry if the The number of times a figure maps onto itself as it figure can be mapped onto itself by a rotation rotates from 0° to 360° is called the order of between 0° and 360° about the center of the figure. symmetry. The wheel has order 10 rotational symmetry. There are 10 bolts and the tire can be The wheel has rotational symmetry. rotated 10 times within 360° and map onto itself.

The wheel has order 8 rotational symmetry and magnitude .

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps 28. onto itself. The wheel has magnitude of symmetry of SOLUTION: . This figure is a square, because each pair of adjacent sides is congruent and perpendicular. ANSWER: All squares have both line and rotational symmetry. yes; 10; 36° The line symmetry is vertically, horizontally, and diagonally through the center of the square, with lines State whether the figure has line symmetry that are either parallel to the sides of the square or and/or rotational symmetry. If so, describe the that include two vertices of the square. The reflections and/or rotations that map the figure equations of those lines are: x = 0, y = 0, y = x, and y onto itself. = -x

The rotational symmetry is for each quarter turn in a square, so the rotations of 90, 180, and 270 degrees around the origin map the square onto itself.

no symmetry the square onto itself.

ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = 0, the reflection in the line y = x, and the reflection in the 30. line y = -x all map the square onto itself; the rotations SOLUTION: of 90, 180, and 270 degrees around the origin map This figure is a parallelogram, so it has rotational

the square onto itself. symmetry of a half turn or 180 degrees around its center, which is the point (1, -1.5).

Since this parallelogram is not a rhombus it does not have line symmetry.

page 263 is a long exposure shot of the Eiffel tower The reflection in the line y = 1.5 maps the trapezoid reflected in a pool. onto itself, because that is the perpendicular bisector

to the parallel sides. a. Describe the two-dimensional symmetry created ANSWER: by the photo. line symmetry; the reflection in the line y = 1.5 maps b. Is there rotational symmetry in the photo? Explain the trapezoid onto itself. your reasoning. SOLUTION: a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no 30. rotational symmetry. SOLUTION: ANSWER: This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its a. Sample answer: There is a horizontal line of center, which is the point (1, -1.5). symmetry between the tower and its reflection. There is a vertical line of symmetry through the Since this parallelogram is not a rhombus it does not center of the photo. have line symmetry. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no ANSWER: rotational symmetry. rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram COORDINATE GEOMETRY Determine onto itself. whether the figure with the given vertices has line symmetry and/or rotational symmetry. 31. MODELING Symmetry is an important component 32. R(–3, 3), S(–3, –3), T(3, 3) of photography. Photographers often use reflection in SOLUTION: water to create symmetry in photos. The photo on page 263 is a long exposure shot of the Eiffel tower Draw the figure on a coordinate plane. reflected in a pool.

a. Describe the two-dimensional symmetry created by the photo. b. Is there rotational symmetry in the photo? Explain your reasoning.

SOLUTION: a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. A figure has line symmetry if the figure can be There is a vertical line of symmetry through the mapped onto itself by a reflection in a line. center of the photo. b No; sample answer: Because of how the image is The given triangle has a line of symmetry through reflected over the horizontal line, there is no points (0, 0) and (–3, 3). rotational symmetry. A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation a. Sample answer: There is a horizontal line of between 0° and 360° about the center of the figure. symmetry between the tower and its reflection. There is not way to rotate the figure and have it map There is a vertical line of symmetry through the onto itself. center of the photo. b No; sample answer: Because of how the image is Thus, the figure has only line symmetry. reflected over the horizontal line, there is no rotational symmetry. ANSWER: line COORDINATE GEOMETRY Determine whether the figure with the given vertices has line symmetry and/or rotational symmetry. 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) 32. R(–3, 3), S(–3, –3), T(3, 3) SOLUTION: SOLUTION: Draw the figure on a coordinate plane. Draw the figure on a coordinate plane.

A figure has line symmetry if the figure can be A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The mapped onto itself by a reflection in a line. given figure has 4 lines of symmetry. The line of symmetry are though the following pairs of points The given triangle has a line of symmetry through {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)},

points (0, 0) and (–3, 3). and {(2, 2), (2, –2)}.

Thus, the figure has both line symmetry and Thus, the figure has only line symmetry. rotational symmetry. ANSWER: ANSWER: 3-5 Symmetryline line and rotational 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 2) SOLUTION: Draw the figure on a coordinate plane. SOLUTION: Draw the figure on a coordinate plane.

A figure has line symmetry if the figure can be A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The mapped onto itself by a reflection in a line. The given figure has 4 lines of symmetry. The line of given hexagon has 2 lines of symmetry. The lines symmetry are though the following pairs of points pass through the following pair of points {(0, 4), (0, – {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, 4)}, and {(3, 0), (–3, 0)} and {(2, 2), (2, –2)}. A figure in the plane has rotational symmetry if the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. between 0° and 360° about the center of the figure. The figure has rotational symmetry. You can rotate The figure can be rotated from the origin and map the figure once within 360° and have it map to itself. onto itself. The order of symmetry is 4.

A figure has line symmetry if the figure can be A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The mapped onto itself by a reflection in a line. The trapezoid has a line of reflection through points (0,3) given hexagon has 2 lines of symmetry. The lines and (0, –3). pass through the following pair of points {(0, 4), (0, – 4)}, and {(3, 0), (–3, 0)} A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation A figure in the plane has rotational symmetry if the eSolutions Manual - Powered by Cognero between 0° and 360° about the center of the Page 14 figure can be mapped onto itself by a rotation figure. There is no way to rotate this figure and have between 0° and 360° about the center of the figure. it map onto itself. Thus, it does not have rotational The figure has rotational symmetry. You can rotate symmetry. the figure once within 360° and have it map to itself. Therefore, the figure has only line symmetry. Thus, the figure has both line symmetry and rotational symmetry. ANSWER: line ANSWER: line and rotational ALGEBRA Graph the function and determine whether the graph has line and/or rotational 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) symmetry. If so, state the order and magnitude of symmetry, and write the equations of any lines of SOLUTION: symmetry. Draw the figure on a coordinate plane. 36. y = x SOLUTION: Graph the function.

symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation Therefore, the figure has only line symmetry. between 0° and 360° about the center of the figure. The line can be rotated twice within 360° and be ANSWER: mapped onto itself. line ALGEBRA Graph the function and determine The number of times a figure maps onto itself as it whether the graph has line and/or rotational rotates from 0° to 360° is called the order of symmetry. If so, state the order and magnitude of symmetry. The graph has order 2 rotational

symmetry, and write the equations of any lines of symmetry. symmetry. 36. y = x The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps SOLUTION: onto itself. Graph the function. The graph has magnitude of symmetry of .

Thus, the graph has both reflectional and rotational symmetry.

ANSWER: rotational; 2; 180°; line symmetry; y = –x

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry.

Thus, the graph has only reflectional symmetry.

ANSWER: line; x = 0

2 37. y = x + 1 SOLUTION: Graph the function.

38. y = –x3 SOLUTION: Graph the function.

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph is reflected through the y-axis. Thus, the equation of the line symmetry is x = 0.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps 38. y = –x3 onto itself. The graph has magnitude of symmetry of SOLUTION: . Graph the function. Thus, the graph has only rotational symmetry.

ANSWER: rotational; 2; 180°

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself. 39. Refer to the rectangle on the coordinate plane. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. You can rotate the graph through the origin and have it map onto itself.

b. The equations of the lines of symmetry do not

change; although the rectangle does not map onto 39. Refer to the rectangle on the coordinate plane. itself under this rotation, the lines of symmetry are mapped to each other.

ANSWER: a. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are

mapped to each other. a. What are the equations of the lines of symmetry of the rectangle? 40. MULTIPLE REPRESENTATIONS In this b. What happens to the equations of the lines of problem, you will use dynamic geometric software to symmetry when the rectangle is rotated 90 degrees investigate line and rotational symmetry in regular counterclockwise around its center of symmetry? polygons. Explain. a. Geometric Use The Geometer’s Sketchpad to SOLUTION: draw an equilateral triangle. Use the reflection tool a. The lines of symmetry are parallel to the sides of under the transformation menu to investigate and the rectangles, and through the center of rotation. determine all possible lines of symmetry. Then record their number. The slopes of the sides of the rectangle are 0.5 and b. Geometric Use the rotation tool under the -2, so the slopes of the lines of symmetry are the transformation menu to investigate the rotational same. symmetry of the figure in part a. Then record its order of symmetry. The center of the rectangle is (1, 1.5). Use the c. Tabular Repeat the process in parts a and b for a point-slope formula to find equations. square, regular pentagon, and regular hexagon.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record Next, draw a line through B perpendicular to . their number. Reflect the triangle in the line. Show the labels of the b. Geometric Use the rotation tool under the reflected image. If the image maps to the original, transformation menu to investigate the rotational then this line is a line of reflection. symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides. SOLUTION: Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the a. Construct an equilateral triangle and label the reflected image. If the image maps to the original, vertices A, B, and C. Draw a line through A then this line is a line of reflection. perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the

reflected image. If the image maps to the original, There are 3 lines of symmetry. then this line is a line of reflection.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

There are 3 lines of symmetry.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines Since the figure maps onto itself 3 times as it is constructed. So there are 5 lines of symmetry. rotated, the order of symmetry is 3. Next, rotate the square about the center point. The c. image maps to the original at 72, 144, 216, 288, and Square 360 degree rotations. So the order of symmetry is 5. Construct a square and then construct lines through the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Regular Hexagon Construct a regular hexagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 6 lines constructed. So there are 6 lines of symmetry. Next, rotate the square about the center point. The Regular Pentagon image maps to the original at 60, 120, 180, 240, 300, Construct a regular pentagon and then construct lines and 360 degree rotations. So the order of symmetry through each vertex perpendicular to the sides. Use is 6. the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5.

d. Sample answer: for each figure studied, the number of sides of the figure is the same as the lines of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and

order of symmetry n. Regular Hexagon Construct a regular hexagon and then construct lines ANSWER: through each vertex perpendicular to the sides. Use a. 3 the reflection tool first to find that the image maps b. 3 onto the original when reflected in each of the 6 lines c. constructed. So there are 6 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 60, 120, 180, 240, 300, and 360 degree rotations. So the order of symmetry is 6. d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

SOLUTION:

A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. This figure has 4 lines of symmetry.

ANSWER: a. 3 A figure in the plane has rotational symmetry if the b. 3 figure can be mapped onto itself by a rotation c. between 0° and 360° about the center of the figure.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

The figure also has rotational symmetry. 41. ERROR ANALYSIS Jaime says that Figure A has

only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them Therefore, neither of them are correct. Figure A has correct? Explain your reasoning. both line and rotational symmetry. ANSWER: Neither; Figure A has both line and rotational symmetry.

CHALLENGE A quadrilateral in the coordinate 42. plane has exactly two lines of symmetry, y = x – 1 SOLUTION: and y = –x + 2. Find a set of possible vertices for A figure has line symmetry if the figure can be the figure. Graph the figure and the lines of mapped onto itself by a reflection in a line. This symmetry. figure has 4 lines of symmetry. SOLUTION: Graph the figure and the lines of symmetry.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Pick points that are the same distance a from one line and the same distance b from the other line. In the same answer, the quadrilateral is a rectangle with sides which are parallel to the lines of symmetry. This guarantees that the vertices of the quadrilateral are the same distance a from one line and the same The figure also has rotational symmetry. distance b from the other line. In this case, a = Therefore, neither of them are correct. Figure A has

both line and rotational symmetry. and b = .

43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. SOLUTION: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines. Pick points that are the same distance a from one line and the same distance b from the other line. In ANSWER: the same answer, the quadrilateral is a rectangle with circle; Every line through the center of a circle is a sides which are parallel to the lines of symmetry. line of symmetry, and there are infinitely many such This guarantees that the vertices of the quadrilateral lines. are the same distance a from one line and the same OPEN-ENDED distance b from the other line. In this case, a = 44. Draw a figure with line symmetry but not rotational symmetry. Explain. and b = . SOLUTION: A figure has line symmetry if the figure can be A set of possible vertices for the figure are, (–1, 0), mapped onto itself by a reflection in a line. A figure (2, 3), (4, 1), and (1, 2). in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and ANSWER: 360° about the center of the figure. Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) Identify a figure that has line symmetry but does not have rotational symmetry.

An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain.

SOLUTION: ANSWER: circle; Every line through the center of a circle is a Sample answer: An isosceles triangle has line line of symmetry, and there are infinitely many such symmetry from the vertex angle to the base of the lines. triangle, but it does not have rotational symmetry ANSWER: because it cannot be rotated from 0° to 360° and

circle; Every line through the center of a circle is a map onto itself. line of symmetry, and there are infinitely many such lines.

44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. SOLUTION: 45. WRITING IN MATH How are line symmetry and A figure has line symmetry if the figure can be rotational symmetry related? mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can SOLUTION: be mapped onto itself by a rotation between 0° and In both types of symmetries the figure is mapped 360° about the center of the figure. onto itself.

Identify a figure that has line symmetry but does not Rotational symmetry. have rotational symmetry.

In some cases an object can have both rotational and reflectional symmetry, such as the diamond, however ANSWER: some objects do not have both such as the crab.

Reflectional symmetry:

A 2 In some cases an object can have both rotational and B 3 reflectional symmetry, such as the diamond, however C some objects do not have both such as the crab. 4 D 8 SOLUTION:

A 2

B 3 C C 4 D 8 SOLUTION: D

2 + 2 = 4, so C is the correct answer. Option B has reflectional symmetry but not rotational ANSWER: symmetry. C

Option D has rotational symmetry but not reflectional

C symmetry.

Option E has reflectional symmetry but not rotational E symmetry.

SOLUTION: Option A has rotational and reflectional symmetry.

The correct choice is D.

Option D has rotational symmetry but not reflectional symmetry. ANSWER: C

symmetry. A B C D SOLUTION: First, plot the points.

The correct choice is D.

ANSWER: D

48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle D Scalene triangle SOLUTION: An isosceles triangle has one line of symmetry and Then, plot each option A-D to consider each figure no rotational symmetry. The correct choice is C. and its symmetry. Option A has both reflectional and rotational symmetry. ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D

SOLUTION: First, plot the points. Option B has reflective symmetry but not rotational symmetry. The correct choice is B.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

Thus, there are four possible lines that go through the center and are lines of reflections.

Therefore, the figure has four lines of symmetry. State whether the figure appears to have line symmetry. Write yes or no. If so, copy the ANSWER: figure, draw all lines of symmetry, and state yes; 4 their number.

SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry. 2. In order for the figure to map onto itself, the line of SOLUTION: reflection must go through the center point. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. Two lines of reflection go through the sides of the figure. The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER: no

Two lines of reflection go through the vertices of the figure. 3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

Thus, there are four possible lines that go through The figure has a vertical line of symmetry. the center and are lines of reflections.

It does not have a horizontal line of symmetry.

Therefore, the figure has four lines of symmetry.

ANSWER: yes; 4 The figure does not have a line of symmetry through the vertices.

Thus, the figure has only one line of symmetry.

ANSWER: 2. yes; 1 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself. State whether the figure has rotational ANSWER: symmetry. Write yes or no. If so, copy the no figure, locate the center of symmetry, and state the order and magnitude of symmetry.

ANSWER: no

It does not have a horizontal line of symmetry.

5. SOLUTION: A figure in the plane has rotational symmetry if the The figure does not have a line of symmetry through figure can be mapped onto itself by a rotation the vertices. between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

Thus, the figure has only one line of symmetry.

ANSWER: yes; 1 The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of symmetry.

The given figure has order of symmetry of 2, since the figure can be rotated twice in 360°. State whether the figure has rotational symmetry. Write yes or no. If so, copy the The magnitude of symmetry is the smallest angle figure, locate the center of symmetry, and state through which a figure can be rotated so that it maps

the order and magnitude of symmetry. onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is .

4. ANSWER: yes; 2; 180° SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry.

ANSWER: 6. no SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

5. The given figure has rotational symmetry. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The number of times a figure maps onto itself as it onto itself. rotates form 0° and 360° is called the order of symmetry. The figure has magnitude of symmetry of . The given figure has order of symmetry of 2, since ANSWER: the figure can be rotated twice in 360°. yes; 4; 90°

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

Since the figure has order 2 rotational symmetry, the magnitude of the symmetry is . ANSWER: State whether the figure has line symmetry yes; 2; 180° and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

6. SOLUTION: Vertical and horizontal lines through the center and SOLUTION: diagonal lines through two opposite vertices are all A figure in the plane has rotational symmetry if the lines of symmetry for a square oriented this way. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The equations of those lines in this figure are x = 0, y = -1, y = x - 1, and y = -x - 1. The given figure has rotational symmetry. Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

Since the figure can be rotated 4 times within 360° , it has order 4 rotational symmetry

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of 8. . SOLUTION: ANSWER: This figure does not have line symmetry, because yes; 4; 90° adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

ANSWER: rotational symmetry; the rotation of 180 degrees around the point (1, 1) maps the parallelogram onto State whether the figure has line symmetry itself. and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure REGULARITY State whether the figure onto itself. appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

9. 7. SOLUTION: SOLUTION: A figure has reflectional symmetry if the figure can Vertical and horizontal lines through the center and be mapped onto itself by a reflection in a line. diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way. For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure The equations of those lines in this figure are x = 0, does not have any lines of of symmetry. y = -1, y = x - 1, and y = -x - 1. ANSWER: Each quarter turn also maps the square onto itself. no So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself.

ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in 10. the line y = -x - 1 map the square onto itself; the SOLUTION: rotations of 90, 180, and 270 degrees around the point A figure has reflectional line symmetry if the figure

(0, -1) map the square onto itself. can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

The figure has a vertical and horizontal line of 8. reflection. SOLUTION: This figure does not have line symmetry, because adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself. It is also possible to have reflection over the ANSWER: diagonal lines. rotational symmetry; the rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

9. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. ANSWER: For the given figure, there are no lines of reflection yes; 4 where the figure can map onto itself. Thus, the figure does not have any lines of of symmetry.

ANSWER: no

10. SOLUTION: A figure has reflectional line symmetry if the figure 11. can be mapped onto itself by a reflection in a line. SOLUTION:

A figure has reflectional symmetry if the figure can

The given figure has reflectional symmetry. be mapped onto itself by a reflection in a line.

In order for the figure to map onto itself, the line of The given hexagon has reflectional symmetry. reflection must go through the center point.

The figure has a vertical and horizontal line of In order for the hexagon to map onto itself, the line of reflection must go through the center point. reflection.

There are three lines of reflection that go though opposites edges.

It is also possible to have reflection over the

diagonal lines. There are three lines of reflection that go though opposites vertices.

Therefore, the figure has four lines of symmetry

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry.

ANSWER: yes; 4

ANSWER: yes; 6

11. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. 12.

SOLUTION: The given hexagon has reflectional symmetry. A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line. In order for the hexagon to map onto itself, the line

of reflection must go through the center point. The figure has reflectional symmetry.

There are three lines of reflection that go though There is only one line of symmetry, a horizontal line opposites edges. through the middle of the figure.

Thus, the figure has one line of symmetry.

There are three lines of reflection that go though ANSWER: opposites vertices. yes; 1

The figure has reflectional symmetry.

There is only one possible line of reflection, horizontally though the middle of the figure.

ANSWER:

yes; 6 Thus, the figure has one line of symmetry.

ANSWER: yes; 1

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their Thus, the figure has one line of symmetry. number. 15. Refer to page 262. ANSWER: SOLUTION: yes; 1 A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

13. ANSWER: SOLUTION: no A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. 16. Refer to the flag on page 262.

SOLUTION: The figure has reflectional symmetry. A figure has reflectional symmetry if the figure can

There is only one possible line of reflection, be mapped onto itself by a reflection in a line.

horizontally though the middle of the figure. The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point. Thus, the figure has one line of symmetry. A horizontal and vertical lines of reflection are ANSWER: possible. yes; 1

14. SOLUTION: A figure has reflectional symmetry if the figure can Two diagonal lines of reflection are possible. be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. It is not possible to draw a line of reflection where the figure can map onto itself.

ANSWER: no

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER: no ANSWER: yes; 4 16. Refer to the flag on page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

A horizontal and vertical lines of reflection are 17. Refer to page 262. possible. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

Two diagonal lines of reflection are possible.

It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

There are a total of four possible lines that go through the center and are lines of reflections. Thus, the flag has four lines of symmetry.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

18.

SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the yes; 4 figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

The number of times a figure maps onto itself as it 17. Refer to page 262. rotates from 0° to 360° is called the order of symmetry. SOLUTION: A figure has reflectional symmetry if the figure can This figure has order 2 rotational symmetry, since be mapped onto itself by a reflection in a line. you have to rotate 180° to get the figure to map onto itself. The figure has reflectional symmetry. The magnitude of symmetry is the smallest angle A horizontal line is a line of reflections for this flag. through which a figure can be rotated so that it maps onto itself.

The figure has a magnitude of symmetry of .

ANSWER: It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry.

ANSWER: yes; 2; 180° yes; 1

The figure has order 3 rotational symmetry.

The figure has rotational symmetry. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The number of times a figure maps onto itself as it onto itself. rotates from 0° to 360° is called the order of symmetry. The figure has magnitude of symmetry of . This figure has order 2 rotational symmetry, since you have to rotate 180° to get the figure to map onto ANSWER: itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. yes; 3; 120° The figure has a magnitude of symmetry of .

ANSWER: 20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation yes; 2; 180° between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

19. ANSWER: SOLUTION: no A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The triangle has rotational symmetry. 21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has order 3 rotational symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 3; 120°

20. SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the no figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself. 22. SOLUTION: ANSWER: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation no between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

21. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER:

yes; 8; 45°

ANSWER: 23. no SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

22. The figure has rotational symmetry. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This means that the figure can be rotated 8 times and map onto itself within 360°.

The magnitude of symmetry is the smallest angle The number of times a figure maps onto itself as it through which a figure can be rotated so that it maps rotates from 0° to 360° is called the order of onto itself. symmetry. The figure has magnitude of symmetry of The figure has order 8 rotational symmetry. This . implies you can rotate the figure 8 times and have it map onto itself within 360°. ANSWER:

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

yes; 8; 45° The figure has magnitude of symmetry of . WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, ANSWER: state the order and magnitude of symmetry. 24. Refer to page 263. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. yes; 8; 45° The wheel has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of

23. symmetry.

SOLUTION: The wheel has order 5 rotational symmetry. There A figure in the plane has rotational symmetry if the are 5 large spokes and 5 small spokes. You can figure can be mapped onto itself by a rotation rotate the wheel 5 times within 360° and map the between 0° and 360° about the center of the figure. figure onto itself.

The figure has rotational symmetry. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The wheel has magnitude of symmetry .

ANSWER: The number of times a figure maps onto itself as it yes; 5; 72° rotates from 0° to 360° is called the order of symmetry. 25. Refer to page 263. The figure has order 8 rotational symmetry. This SOLUTION: means that the figure can be rotated 8 times and map A figure in the plane has rotational symmetry if the

onto itself within 360°. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The wheel has rotational symmetry.

onto itself.

The number of times a figure maps onto itself as it The figure has magnitude of symmetry of rotates from 0° to 360° is called the order of . symmetry. ANSWER: The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle yes; 8; 45° through which a figure can be rotated so that it maps onto itself. WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, The wheel has order 8 rotational symmetry and state the order and magnitude of symmetry. magnitude . 24. Refer to page 263. ANSWER: SOLUTION: yes; 8; 45° A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 26. Refer to page 263. between 0° and 360° about the center of the figure. SOLUTION: The wheel has rotational symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The number of times a figure maps onto itself as it between 0° and 360° about the center of the figure. rotates from 0° to 360° is called the order of The wheel has rotational symmetry. symmetry. The number of times a figure maps onto itself as it The wheel has order 5 rotational symmetry. There rotates from 0° to 360° is called the order of are 5 large spokes and 5 small spokes. You can symmetry. The wheel has order 10 rotational rotate the wheel 5 times within 360° and map the symmetry. There are 10 bolts and the tire can be figure onto itself. rotated 10 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps through which a figure can be rotated so that it maps onto itself. onto itself. The wheel has magnitude of symmetry of . The wheel has magnitude of symmetry ANSWER: . yes; 10; 36° ANSWER: State whether the figure has line symmetry yes; 5; 72° and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure 25. Refer to page 263. onto itself.

SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The wheel has rotational symmetry. 27. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of SOLUTION: symmetry. This triangle is scalene, so it cannot have symmetry.

The wheel has order 8 rotational symmetry. There ANSWER: are 8 spokes, thus the wheel can be rotated 8 times no symmetry within 360° and map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The wheel has order 8 rotational symmetry and magnitude . 28.

ANSWER: SOLUTION: yes; 8; 45° This figure is a square, because each pair of adjacent sides is congruent and perpendicular. 26. Refer to page 263. All squares have both line and rotational symmetry. The line symmetry is vertically, horizontally, and SOLUTION: diagonally through the center of the square, with lines A figure in the plane has rotational symmetry if the that are either parallel to the sides of the square or figure can be mapped onto itself by a rotation that include two vertices of the square. The between 0° and 360° about the center of the figure. equations of those lines are: x = 0, y = 0, y = x, and y The wheel has rotational symmetry. = -x

The number of times a figure maps onto itself as it The rotational symmetry is for each quarter turn in a rotates from 0° to 360° is called the order of square, so the rotations of 90, 180, and 270 degrees symmetry. The wheel has order 10 rotational around the origin map the square onto itself. symmetry. There are 10 bolts and the tire can be rotated 10 times within 360° and map onto itself. ANSWER: The magnitude of symmetry is the smallest angle line symmetry; rotational symmetry; the reflection in through which a figure can be rotated so that it maps the line x = 0, the reflection in the line y = 0, the onto itself. The wheel has magnitude of symmetry of reflection in the line y = x, and the reflection in the . line y = -x all map the square onto itself; the rotations of 90, 180, and 270 degrees around the origin map ANSWER: the square onto itself. yes; 10; 36° State whether the figure has line symmetry and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself.

29. SOLUTION: The trapezoid has line symmetry, because it is isosceles, but it does not have rotational symmetry, because no trapezoid does. 27. The reflection in the line y = 1.5 maps the trapezoid SOLUTION: onto itself, because that is the perpendicular bisector This triangle is scalene, so it cannot have symmetry. to the parallel sides.

ANSWER: ANSWER: no symmetry line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself.

28. 30. SOLUTION: This figure is a square, because each pair of adjacent SOLUTION: sides is congruent and perpendicular. This figure is a parallelogram, so it has rotational All squares have both line and rotational symmetry. symmetry of a half turn or 180 degrees around its The line symmetry is vertically, horizontally, and center, which is the point (1, -1.5). diagonally through the center of the square, with lines that are either parallel to the sides of the square or Since this parallelogram is not a rhombus it does not that include two vertices of the square. The have line symmetry. equations of those lines are: x = 0, y = 0, y = x, and y = -x ANSWER: rotational symmetry; the rotation of 180 degrees The rotational symmetry is for each quarter turn in a around the point (1, -1.5) maps the parallelogram square, so the rotations of 90, 180, and 270 degrees onto itself. around the origin map the square onto itself. 31. MODELING Symmetry is an important component of photography. Photographers often use reflection in ANSWER: water to create symmetry in photos. The photo on line symmetry; rotational symmetry; the reflection in page 263 is a long exposure shot of the Eiffel tower the line x = 0, the reflection in the line y = 0, the reflected in a pool. reflection in the line y = x, and the reflection in the line y = -x all map the square onto itself; the rotations a. Describe the two-dimensional symmetry created of 90, 180, and 270 degrees around the origin map by the photo. the square onto itself. b. Is there rotational symmetry in the photo? Explain your reasoning. SOLUTION: a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. 29. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no SOLUTION: rotational symmetry. The trapezoid has line symmetry, because it is isosceles, but it does not have rotational symmetry, ANSWER: because no trapezoid does. a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. The reflection in the line y = 1.5 maps the trapezoid There is a vertical line of symmetry through the onto itself, because that is the perpendicular bisector center of the photo. to the parallel sides. b No; sample answer: Because of how the image is ANSWER: reflected over the horizontal line, there is no rotational symmetry. line symmetry; the reflection in the line y = 1.5 maps the trapezoid onto itself. COORDINATE GEOMETRY Determine whether the figure with the given vertices has line symmetry and/or rotational symmetry. 32. R(–3, 3), S(–3, –3), T(3, 3) SOLUTION: Draw the figure on a coordinate plane.

30. SOLUTION: This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its center, which is the point (1, -1.5).

Since this parallelogram is not a rhombus it does not have line symmetry. A figure has line symmetry if the figure can be ANSWER: mapped onto itself by a reflection in a line. rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram The given triangle has a line of symmetry through onto itself. points (0, 0) and (–3, 3).

a. Describe the two-dimensional symmetry created Thus, the figure has only line symmetry. by the photo. b. Is there rotational symmetry in the photo? Explain ANSWER: your reasoning. line SOLUTION: 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) a Sample answer: There is a horizontal line of symmetry between the tower and its reflection. SOLUTION: There is a vertical line of symmetry through the Draw the figure on a coordinate plane. center of the photo. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no rotational symmetry.

ANSWER: a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the

center of the photo. A figure has line symmetry if the figure can be b No; sample answer: Because of how the image is mapped onto itself by a reflection in a line. The reflected over the horizontal line, there is no given figure has 4 lines of symmetry. The line of rotational symmetry. symmetry are though the following pairs of points COORDINATE GEOMETRY Determine {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)},

whether the figure with the given vertices has line and {(2, 2), (2, –2)}. symmetry and/or rotational symmetry. 32. R(–3, 3), S(–3, –3), T(3, 3) A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. Draw the figure on a coordinate plane. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational

34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 2) A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. SOLUTION: Draw the figure on a coordinate plane. The given triangle has a line of symmetry through points (0, 0) and (–3, 3).

A figure has line symmetry if the figure can be Thus, the figure has only line symmetry. mapped onto itself by a reflection in a line. The ANSWER: given hexagon has 2 lines of symmetry. The lines pass through the following pair of points {(0, 4), (0, – line 4)}, and {(3, 0), (–3, 0)}

33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. Draw the figure on a coordinate plane. The figure has rotational symmetry. You can rotate the figure once within 360° and have it map to itself.

Thus, the figure has both line symmetry and rotational symmetry.

ANSWER: line and rotational

35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The SOLUTION: given figure has 4 lines of symmetry. The line of Draw the figure on a coordinate plane. symmetry are though the following pairs of points {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, and {(2, 2), (2, –2)}.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4. A figure has line symmetry if the figure can be Thus, the figure has both line symmetry and mapped onto itself by a reflection in a line. The rotational symmetry. trapezoid has a line of reflection through points (0,3) and (0, –3). ANSWER: line and rotational A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – between 0° and 360° about the center of the

2) figure. There is no way to rotate this figure and have SOLUTION: it map onto itself. Thus, it does not have rotational Draw the figure on a coordinate plane. symmetry.

Therefore, the figure has only line symmetry.

ANSWER: line

ALGEBRA Graph the function and determine whether the graph has line and/or rotational symmetry. If so, state the order and magnitude of A figure has line symmetry if the figure can be symmetry, and write the equations of any lines of mapped onto itself by a reflection in a line. The symmetry. given hexagon has 2 lines of symmetry. The lines 36. y = x pass through the following pair of points {(0, 4), (0, – SOLUTION: 4)}, and {(3, 0), (–3, 0)} Graph the function. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The figure has rotational symmetry. You can rotate the figure once within 360° and have it map to itself.

Thus, the figure has both line symmetry and rotational symmetry.

3-5 SymmetryANSWER: line and rotational A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) line y = x has reflectional symmetry since any line SOLUTION: perpendicular to y = x is a line of reflection. The equation of the line symmetry is y = –x. Draw the figure on a coordinate plane.

Therefore, the figure has only line symmetry. rotational; 2; 180°; line symmetry; y = –x ANSWER: line

A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line. The line y = x has reflectional symmetry since any line A figure has reflectional symmetry if the figure can eSolutions Manual - Powered by Cognero Page 15 perpendicular to y = x is a line of reflection. The be mapped onto itself by a reflection in a line. The equation of the line symmetry is y = –x. graph is reflected through the y-axis. Thus, the equation of the line symmetry is x = 0. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation A figure in the plane has rotational symmetry if the between 0° and 360° about the center of the figure. figure can be mapped onto itself by a rotation The line can be rotated twice within 360° and be between 0° and 360° about the center of the mapped onto itself. figure. There is no way to rotate the graph and have it map onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of Thus, the graph has only reflectional symmetry. symmetry. The graph has order 2 rotational symmetry. ANSWER: line; x = 0 The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of .

Thus, the graph has both reflectional and rotational symmetry.

ANSWER: rotational; 2; 180°; line symmetry; y = –x 38. y = –x3 SOLUTION: Graph the function.

2 37. y = x + 1 SOLUTION: A figure has reflectional symmetry if the figure can Graph the function. be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself.

39. Refer to the rectangle on the coordinate plane. 38. y = –x3 SOLUTION: Graph the function.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record 39. Refer to the rectangle on the coordinate plane. their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. a. What are the equations of the lines of symmetry of d. Verbal Make a conjecture about the number of the rectangle? lines of symmetry and the order of symmetry for a b. What happens to the equations of the lines of regular polygon with n sides. symmetry when the rectangle is rotated 90 degrees SOLUTION: counterclockwise around its center of symmetry? a. Construct an equilateral triangle and label the Explain. vertices A, B, and C. Draw a line through A SOLUTION: perpendicular to . Reflect the triangle in the line. a. The lines of symmetry are parallel to the sides of Show the labels of the reflected image. If the image the rectangles, and through the center of rotation. maps to the original, then this line is a line of reflection. The slopes of the sides of the rectangle are 0.5 and -2, so the slopes of the lines of symmetry are the same. The center of the rectangle is (1, 1.5). Use the point-slope formula to find equations.

b. The equations of the lines of symmetry do not change; although the rectangle does not map onto Next, draw a line through B perpendicular to . itself under this rotation, the lines of symmetry are Reflect the triangle in the line. Show the labels of the mapped to each other. reflected image. If the image maps to the original, then this line is a line of reflection. ANSWER: a. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular Lastly, draw a line through C perpendicular to . polygons. Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, a. Geometric Use The Geometer’s Sketchpad to then this line is a line of reflection. draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the order of symmetry for each polygon. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides. SOLUTION:

a. Construct an equilateral triangle and label the There are 3 lines of symmetry. vertices A, B, and C. Draw a line through A perpendicular to . Reflect the triangle in the line. b. Construct an equilateral triangle and show the Show the labels of the reflected image. If the image labels of the vertices. Next, find the center of the maps to the original, then this line is a line of triangle. Since this is an equilateral triangle, the reflection. circumcenter, incenter, centroid, and orthocenter are the same point. Construct altitudes through each vertex and label the intersection. Rotate the triangle about point D. A 120 degree rotation will map the image to the original. Show the labels of the image.

Next, draw a line through B perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

There are 3 lines of symmetry.

b. Construct an equilateral triangle and show the labels of the vertices. Next, find the center of the triangle. Since this is an equilateral triangle, the circumcenter, incenter, centroid, and orthocenter are Since the figure maps onto itself 3 times as it is the same point. Construct altitudes through each rotated, the order of symmetry is 3. vertex and label the intersection. c. Rotate the triangle about point D. A 120 degree Square rotation will map the image to the original. Show the Construct a square and then construct lines through labels of the image. the midpoints of each side and diagonals. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 4 lines constructed. So there are 4 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and The triangle can be rotated a third time about D. A 360 degree rotations. So the order of symmetry is 5. 360 degree rotation maps the image to the original.

Regular Pentagon Construct a regular pentagon and then construct lines d. Sample answer: for each figure studied, the through each vertex perpendicular to the sides. Use number of sides of the figure is the same as the lines the reflection tool first to find that the image maps of symmetry and the order of symmetry. A regular onto the original when reflected in each of the 5 lines polygon with n sides has n lines of symmetry and constructed. So there are 5 lines of symmetry. order of symmetry n. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and ANSWER: 360 degree rotations. So the order of symmetry is 5. a. 3 b. 3 c.

d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure also has rotational symmetry.

Therefore, neither of them are correct. Figure A has d. Sample answer: for each figure studied, the both line and rotational symmetry. number of sides of the figure is the same as the lines of symmetry and the order of symmetry. A regular ANSWER: polygon with n sides has n lines of symmetry and Neither; Figure A has both line and rotational order of symmetry n. symmetry.

ANSWER: 42. CHALLENGE A quadrilateral in the coordinate a. 3 plane has exactly two lines of symmetry, y = x – 1 b. 3 and y = –x + 2. Find a set of possible vertices for c. the figure. Graph the figure and the lines of symmetry. SOLUTION: Graph the figure and the lines of symmetry. d. Sample answer: A regular polygon with n sides has n lines of symmetry and order of symmetry n.

41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A has only rotational symmetry. Is either of them correct? Explain your reasoning.

Pick points that are the same distance a from one line and the same distance b from the other line. In the same answer, the quadrilateral is a rectangle with sides which are parallel to the lines of symmetry. SOLUTION: This guarantees that the vertices of the quadrilateral A figure has line symmetry if the figure can be are the same distance a from one line and the same mapped onto itself by a reflection in a line. This distance b from the other line. In this case, a = figure has 4 lines of symmetry. and b = .

A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2).

ANSWER: Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure also has rotational symmetry. 43. REASONING A figure has infinitely many lines of symmetry. What is the figure? Explain. Therefore, neither of them are correct. Figure A has both line and rotational symmetry. SOLUTION: circle; Every line through the center of a circle is a ANSWER: line of symmetry, and there are infinitely many such Neither; Figure A has both line and rotational lines. symmetry. ANSWER: 42. CHALLENGE A quadrilateral in the coordinate circle; Every line through the center of a circle is a plane has exactly two lines of symmetry, y = x – 1 line of symmetry, and there are infinitely many such and y = –x + 2. Find a set of possible vertices for lines. the figure. Graph the figure and the lines of symmetry. 44. OPEN-ENDED Draw a figure with line symmetry but not rotational symmetry. Explain. SOLUTION: SOLUTION: Graph the figure and the lines of symmetry. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not have rotational symmetry.

Pick points that are the same distance a from one An isosceles triangle has line symmetry from the line and the same distance b from the other line. In vertex angle to the base of the triangle, but it does the same answer, the quadrilateral is a rectangle with not have rotational symmetry because it cannot be

sides which are parallel to the lines of symmetry. rotated from 0° to 360° and map onto itself. This guarantees that the vertices of the quadrilateral are the same distance a from one line and the same distance b from the other line. In this case, a =

and b = . ANSWER: A set of possible vertices for the figure are, (–1, 0), Sample answer: An isosceles triangle has line (2, 3), (4, 1), and (1, 2). symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry ANSWER: because it cannot be rotated from 0° to 360° and Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) map onto itself.

45. WRITING IN MATH How are line symmetry and rotational symmetry related? SOLUTION: 43. REASONING A figure has infinitely many lines of In both types of symmetries the figure is mapped symmetry. What is the figure? Explain. onto itself.

SOLUTION: Rotational symmetry. circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such

lines. Reflectional symmetry: ANSWER: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines.

OPEN-ENDED 44. Draw a figure with line symmetry but not rotational symmetry. Explain. In some cases an object can have both rotational and SOLUTION: reflectional symmetry, such as the diamond, however some objects do not have both such as the crab. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not have rotational symmetry.

An isosceles triangle has line symmetry from the ANSWER: vertex angle to the base of the triangle, but it does Sample answer: In both rotational and line symmetry not have rotational symmetry because it cannot be a figure is mapped onto itself. However, in line rotated from 0° to 360° and map onto itself. symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry.

46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of ANSWER: symmetry and the order of symmetry, and then she Sample answer: An isosceles triangle has line enters this value into a database. Which value should symmetry from the vertex angle to the base of the she enter in the database for the tile shown here? triangle, but it does not have rotational symmetry because it cannot be rotated from 0° to 360° and map onto itself.

A 2 B 3 45. WRITING IN MATH How are line symmetry and C 4 rotational symmetry related? D 8 SOLUTION: SOLUTION: In both types of symmetries the figure is mapped onto itself.

Rotational symmetry.

The tile is a rhombus and has 2 lines of symmetry. Reflectional symmetry: Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn around its center.

2 + 2 = 4, so C is the correct answer.

In some cases an object can have both rotational and ANSWER: reflectional symmetry, such as the diamond, however C some objects do not have both such as the crab. 47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A

ANSWER: Sample answer: In both rotational and line symmetry a figure is mapped onto itself. However, in line C symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have D line symmetry and rotational symmetry.

46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of E symmetry and the order of symmetry, and then she enters this value into a database. Which value should she enter in the database for the tile shown here? SOLUTION: Option A has rotational and reflectional symmetry.

Option B has reflectional symmetry but not rotational A 2 symmetry. B 3 C 4 D 8 SOLUTION:

Option C has neither rotational nor reflectional symmetry.

The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile. Option D has rotational symmetry but not reflectional It has an order of symmetry of 2, because it has symmetry. rotational symmetry at 180 degrees, or each half turn around its center.

2 + 2 = 4, so C is the correct answer.

ANSWER: Option E has reflectional symmetry but not rotational C symmetry.

47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? A

B The correct choice is D. ANSWER: D

C 48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle D B Equiangular triangle C Isosceles triangle D Scalene triangle

E SOLUTION: An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C. SOLUTION: Option A has rotational and reflectional symmetry. ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points

Option B has reflectional symmetry but not rotational can she plot so that the resulting quadrilateral PQRS symmetry. has line symmetry but not rotational symmetry?

A B C D

Option C has neither rotational nor reflectional SOLUTION: symmetry. First, plot the points.

Option D has rotational symmetry but not reflectional symmetry.

Option E has reflectional symmetry but not rotational symmetry.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

The correct choice is D.

ANSWER: D

48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle D Scalene triangle Option B has reflective symmetry but not rotational SOLUTION: symmetry. The correct choice is B. An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry?

A B C D SOLUTION: First, plot the points.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

Thus, there are four possible lines that go through State whether the figure appears to have line symmetry. Write yes or no. If so, copy the the center and are lines of reflections. figure, draw all lines of symmetry, and state their number.

SOLUTION: A figure has reflectional symmetry if the figure can Therefore, the figure has four lines of symmetry. be mapped onto itself by a reflection in a line. ANSWER: The figure has reflectional symmetry. yes; 4

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

SOLUTION: Two lines of reflection go through the vertices of the A figure has reflectional symmetry if the figure can figure. be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.

ANSWER: no

Thus, there are four possible lines that go through the center and are lines of reflections.

3. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

Therefore, the figure has four lines of symmetry. The figure has a vertical line of symmetry. ANSWER: yes; 4

It does not have a horizontal line of symmetry.

The figure does not have a line of symmetry through 2. the vertices. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself. ANSWER: Thus, the figure has only one line of symmetry. no ANSWER: yes; 1

3. SOLUTION: A figure has reflectional symmetry if the figure can

be mapped onto itself by a reflection in a line. State whether the figure has rotational symmetry. Write yes or no. If so, copy the The given figure has reflectional symmetry. figure, locate the center of symmetry, and state the order and magnitude of symmetry. The figure has a vertical line of symmetry.

4. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation It does not have a horizontal line of symmetry. between 0° and 360° about the center of the figure.

For the given figure, there is no rotation between 0° and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational symmetry. The figure does not have a line of symmetry through the vertices. ANSWER: no

5. Thus, the figure has only one line of symmetry. SOLUTION: A figure in the plane has rotational symmetry if the ANSWER: figure can be mapped onto itself by a rotation yes; 1 between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of symmetry. 4. SOLUTION: The given figure has order of symmetry of 2, since A figure in the plane has rotational symmetry if the the figure can be rotated twice in 360°. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps For the given figure, there is no rotation between 0° onto itself. and 360° that maps the figure onto itself. If the figure were a regular pentagon, it would have rotational Since the figure has order 2 rotational symmetry, the symmetry. magnitude of the symmetry is .

ANSWER: ANSWER: no yes; 2; 180°

5. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. 6.

The given figure has rotational symmetry. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates form 0° and 360° is called the order of

symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The given figure has order of symmetry of 2, since symmetry. the figure can be rotated twice in 360°. Since the figure can be rotated 4 times within 360° , The magnitude of symmetry is the smallest angle it has order 4 rotational symmetry through which a figure can be rotated so that it maps onto itself. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps Since the figure has order 2 rotational symmetry, the onto itself. magnitude of the symmetry is . The figure has magnitude of symmetry of ANSWER: . yes; 2; 180° ANSWER: yes; 4; 90°

6. State whether the figure has line symmetry and/or rotational symmetry. If so, describe the SOLUTION: reflections and/or rotations that map the figure A figure in the plane has rotational symmetry if the onto itself. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The given figure has rotational symmetry.

7. SOLUTION: Vertical and horizontal lines through the center and diagonal lines through two opposite vertices are all lines of symmetry for a square oriented this way. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The equations of those lines in this figure are x = 0, symmetry. y = -1, y = x - 1, and y = -x - 1.

Since the figure can be rotated 4 times within 360° , Each quarter turn also maps the square onto itself. it has order 4 rotational symmetry So the rotations of 90, 180, and 270 degrees around the point (0, -1) map the square onto itself. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps ANSWER: onto itself. line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = -1, the The figure has magnitude of symmetry of reflection in the line y = x - 1, and the reflection in . the line y = -x - 1 map the square onto itself; the rotations of 90, 180, and 270 degrees around the point ANSWER: (0, -1) map the square onto itself. yes; 4; 90°

8. State whether the figure has line symmetry and/or rotational symmetry. If so, describe the SOLUTION: reflections and/or rotations that map the figure This figure does not have line symmetry, because onto itself. adjacent sides are not congruent.

It does have rotational symmetry for each half turn around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself.

ANSWER: rotational symmetry; the rotation of 180 degrees 7. around the point (1, 1) maps the parallelogram onto SOLUTION: itself. Vertical and horizontal lines through the center and REGULARITY State whether the figure diagonal lines through two opposite vertices are all appears to have line symmetry. Write yes or no. lines of symmetry for a square oriented this way. If so, copy the figure, draw all lines of The equations of those lines in this figure are x = 0, symmetry, and state their number. y = -1, y = x - 1, and y = -x - 1.

Each quarter turn also maps the square onto itself. So the rotations of 90, 180, and 270 degrees around 9. the point (0, -1) map the square onto itself. SOLUTION: ANSWER: A figure has reflectional symmetry if the figure can line symmetry; rotational symmetry; the reflection in be mapped onto itself by a reflection in a line. the line x = 0, the reflection in the line y = -1, the reflection in the line y = x - 1, and the reflection in For the given figure, there are no lines of reflection the line y = -x - 1 map the square onto itself; the where the figure can map onto itself. Thus, the figure rotations of 90, 180, and 270 degrees around the point does not have any lines of of symmetry. (0, -1) map the square onto itself. ANSWER: no

10. 8. SOLUTION: SOLUTION: A figure has reflectional line symmetry if the figure This figure does not have line symmetry, because can be mapped onto itself by a reflection in a line. adjacent sides are not congruent.

It does have rotational symmetry for each half turn The given figure has reflectional symmetry. around its center, so a rotation of 180 degrees around the point (1, 1) maps the parallelogram onto itself. In order for the figure to map onto itself, the line of reflection must go through the center point. ANSWER: rotational symmetry; the rotation of 180 degrees The figure has a vertical and horizontal line of around the point (1, 1) maps the parallelogram onto reflection. itself.

REGULARITY State whether the figure appears to have line symmetry. Write yes or no. If so, copy the figure, draw all lines of symmetry, and state their number.

It is also possible to have reflection over the diagonal lines. 9. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

For the given figure, there are no lines of reflection where the figure can map onto itself. Thus, the figure Therefore, the figure has four lines of symmetry does not have any lines of of symmetry.

ANSWER: no

ANSWER: 10. yes; 4 SOLUTION: A figure has reflectional line symmetry if the figure can be mapped onto itself by a reflection in a line.

The given figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

The figure has a vertical and horizontal line of reflection.

11. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

It is also possible to have reflection over the The given hexagon has reflectional symmetry. diagonal lines. In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though opposites edges.

Therefore, the figure has four lines of symmetry

There are three lines of reflection that go though opposites vertices.

ANSWER: yes; 4

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has six lines of symmetry.

11.

SOLUTION: A figure has reflectional symmetry if the figure can ANSWER: be mapped onto itself by a reflection in a line. yes; 6

The given hexagon has reflectional symmetry.

In order for the hexagon to map onto itself, the line of reflection must go through the center point.

There are three lines of reflection that go though opposites edges.

12. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. There are three lines of reflection that go though opposites vertices. The figure has reflectional symmetry.

There is only one line of symmetry, a horizontal line through the middle of the figure.

There are six possible lines that go through the center and are lines of reflections. Thus, the hexagon has Thus, the figure has one line of symmetry.

six lines of symmetry. ANSWER: yes; 1

ANSWER: 13. yes; 6 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

There is only one possible line of reflection, horizontally though the middle of the figure.

12. Thus, the figure has one line of symmetry.

SOLUTION: ANSWER: A figure has reflectional symmetry if the figure can yes; 1 be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

There is only one line of symmetry, a horizontal line through the middle of the figure.

14. SOLUTION: A figure has reflectional symmetry if the figure can Thus, the figure has one line of symmetry. be mapped onto itself by a reflection in a line.

ANSWER: The given figure does not have reflectional yes; 1 symmetry. It is not possible to draw a line of reflection where the figure can map onto itself.

ANSWER: no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the 13. flag, draw all lines of symmetry, and state their number. SOLUTION: 15. Refer to page 262. A figure has reflectional symmetry if the figure can SOLUTION: be mapped onto itself by a reflection in a line. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The figure has reflectional symmetry.

The flag does not have any reflectional symmetry. If There is only one possible line of reflection, the red lines in the diagonals were in the same horizontally though the middle of the figure. location above and below the center horizontal line, the flag would have three lines of symmetry.

ANSWER:

Thus, the figure has one line of symmetry. no

ANSWER: 16. Refer to the flag on page 262. yes; 1 SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of 14. reflection must go through the center point.

SOLUTION: A horizontal and vertical lines of reflection are A figure has reflectional symmetry if the figure can possible. be mapped onto itself by a reflection in a line.

The given figure does not have reflectional symmetry. It is not possible to draw a line of reflection where the figure can map onto itself.

ANSWER: no

FLAGS State whether each flag design appears to have line symmetry. Write yes or no. If so, copy the flag, draw all lines of symmetry, and state their Two diagonal lines of reflection are possible. number. 15. Refer to page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The flag does not have any reflectional symmetry. If the red lines in the diagonals were in the same location above and below the center horizontal line,

the flag would have three lines of symmetry. There are a total of four possible lines that go ANSWER: through the center and are lines of reflections. Thus, the flag has four lines of symmetry. no

16. Refer to the flag on page 262. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point. ANSWER:

yes; 4 A horizontal and vertical lines of reflection are possible.

17. Refer to page 262. Two diagonal lines of reflection are possible. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag.

There are a total of four possible lines that go through the center and are lines of reflections. Thus, the flag has four lines of symmetry. It is not possible to reflect over a vertical or line through the diagonals.

Thus, the figure has one line of symmetry.

ANSWER: yes; 1

ANSWER: yes; 4 State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry.

18. SOLUTION:

A figure in the plane has rotational symmetry if the 17. Refer to page 262. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

A horizontal line is a line of reflections for this flag. The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

It is not possible to reflect over a vertical or line This figure has order 2 rotational symmetry, since through the diagonals. you have to rotate 180° to get the figure to map onto

itself.

Thus, the figure has one line of symmetry. The magnitude of symmetry is the smallest angle ANSWER: through which a figure can be rotated so that it maps onto itself. yes; 1

The figure has a magnitude of symmetry of .

ANSWER:

State whether the figure has rotational symmetry. Write yes or no. If so, copy the figure, locate the center of symmetry, and state the order and magnitude of symmetry. yes; 2; 180°

18. SOLUTION: 19. A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation A figure in the plane has rotational symmetry if the between 0° and 360° about the center of the figure. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The triangle has rotational symmetry.

The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The number of times a figure maps onto itself as it This figure has order 2 rotational symmetry, since rotates from 0° to 360° is called the order of you have to rotate 180° to get the figure to map onto symmetry. itself.

The figure has order 3 rotational symmetry. The magnitude of symmetry is the smallest angle

through which a figure can be rotated so that it maps onto itself. The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The figure has a magnitude of symmetry of

. The figure has magnitude of symmetry of ANSWER: . ANSWER:

yes; 2; 180°

yes; 3; 120°

19. SOLUTION: 20. A figure in the plane has rotational symmetry if the SOLUTION: figure can be mapped onto itself by a rotation A figure in the plane has rotational symmetry if the between 0° and 360° about the center of the figure. figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The triangle has rotational symmetry. The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

ANSWER: no

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. 21. The figure has order 3 rotational symmetry. SOLUTION: The magnitude of symmetry is the smallest angle A figure in the plane has rotational symmetry if the through which a figure can be rotated so that it maps figure can be mapped onto itself by a rotation onto itself. between 0° and 360° about the center of the figure.

The figure has magnitude of symmetry of The crescent shaped figure has no rotational . symmetry. There is no way to rotate it such that it can be mapped onto itself. ANSWER:

yes; 3; 120°

20. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The isosceles trapezoid has no rotational symmetry. There is no way to rotate it such that it can be mapped onto itself.

ANSWER:

no ANSWER: no

21. SOLUTION: 22. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation SOLUTION: between 0° and 360° about the center of the figure. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation The crescent shaped figure has no rotational between 0° and 360° about the center of the figure. symmetry. There is no way to rotate it such that it can be mapped onto itself. The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This implies you can rotate the figure 8 times and have it

map onto itself within 360°.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself.

The figure has magnitude of symmetry of .

ANSWER: ANSWER: no

yes; 8; 45° 22. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. 23.

The figure has rotational symmetry. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

The figure has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The number of times a figure maps onto itself as it The figure has order 8 rotational symmetry. This rotates from 0° to 360° is called the order of

implies you can rotate the figure 8 times and have it symmetry. map onto itself within 360°. The figure has order 8 rotational symmetry. This The magnitude of symmetry is the smallest angle means that the figure can be rotated 8 times and map

through which a figure can be rotated so that it maps onto itself within 360°. onto itself. The magnitude of symmetry is the smallest angle The figure has magnitude of symmetry of through which a figure can be rotated so that it maps

. onto itself.

ANSWER: The figure has magnitude of symmetry of .

ANSWER:

yes; 8; 45° yes; 8; 45°

WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so,

23. state the order and magnitude of symmetry. 24. Refer to page 263. SOLUTION: SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation A figure in the plane has rotational symmetry if the between 0° and 360° about the center of the figure. figure can be mapped onto itself by a rotation

between 0° and 360° about the center of the figure.

The figure has rotational symmetry. The wheel has rotational symmetry.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry.

The figure has order 8 rotational symmetry. This The magnitude of symmetry is the smallest angle means that the figure can be rotated 8 times and map through which a figure can be rotated so that it maps onto itself within 360°. onto itself.

The magnitude of symmetry is the smallest angle The wheel has magnitude of symmetry through which a figure can be rotated so that it maps . onto itself. ANSWER: The figure has magnitude of symmetry of yes; 5; 72° . 25. Refer to page 263. ANSWER: SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

yes; 8; 45° The wheel has rotational symmetry. WHEELS State whether each wheel cover appears to have rotational symmetry. Write yes or no. If so, The number of times a figure maps onto itself as it state the order and magnitude of symmetry. rotates from 0° to 360° is called the order of 24. Refer to page 263. symmetry.

SOLUTION: The wheel has order 8 rotational symmetry. There A figure in the plane has rotational symmetry if the are 8 spokes, thus the wheel can be rotated 8 times figure can be mapped onto itself by a rotation within 360° and map onto itself. between 0° and 360° about the center of the figure. The magnitude of symmetry is the smallest angle The wheel has rotational symmetry. through which a figure can be rotated so that it maps onto itself. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of The wheel has order 8 rotational symmetry and symmetry. magnitude .

The wheel has order 5 rotational symmetry. There ANSWER: are 5 large spokes and 5 small spokes. You can yes; 8; 45° rotate the wheel 5 times within 360° and map the figure onto itself. 26. Refer to page 263. SOLUTION: The magnitude of symmetry is the smallest angle A figure in the plane has rotational symmetry if the through which a figure can be rotated so that it maps figure can be mapped onto itself by a rotation onto itself. between 0° and 360° about the center of the figure.

The wheel has rotational symmetry. The wheel has magnitude of symmetry . The number of times a figure maps onto itself as it ANSWER: rotates from 0° to 360° is called the order of symmetry. The wheel has order 10 rotational yes; 5; 72° symmetry. There are 10 bolts and the tire can be rotated 10 times within 360° and map onto itself. 25. Refer to page 263. SOLUTION: The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps A figure in the plane has rotational symmetry if the onto itself. The wheel has magnitude of symmetry of figure can be mapped onto itself by a rotation . between 0° and 360° about the center of the figure. ANSWER: The wheel has rotational symmetry. yes; 10; 36°

The number of times a figure maps onto itself as it State whether the figure has line symmetry rotates from 0° to 360° is called the order of and/or rotational symmetry. If so, describe the symmetry. reflections and/or rotations that map the figure onto itself. The wheel has order 8 rotational symmetry. There are 8 spokes, thus the wheel can be rotated 8 times within 360° and map onto itself.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. 27. The wheel has order 8 rotational symmetry and SOLUTION: magnitude . This triangle is scalene, so it cannot have symmetry. ANSWER: ANSWER: yes; 8; 45° no symmetry 26. Refer to page 263. SOLUTION: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the figure. The wheel has rotational symmetry. 28. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of SOLUTION: symmetry. The wheel has order 10 rotational This figure is a square, because each pair of adjacent symmetry. There are 10 bolts and the tire can be sides is congruent and perpendicular. rotated 10 times within 360° and map onto itself. All squares have both line and rotational symmetry. The line symmetry is vertically, horizontally, and The magnitude of symmetry is the smallest angle diagonally through the center of the square, with lines through which a figure can be rotated so that it maps that are either parallel to the sides of the square or onto itself. The wheel has magnitude of symmetry of that include two vertices of the square. The . equations of those lines are: x = 0, y = 0, y = x, and y = -x ANSWER: yes; 10; 36° The rotational symmetry is for each quarter turn in a square, so the rotations of 90, 180, and 270 degrees State whether the figure has line symmetry around the origin map the square onto itself. and/or rotational symmetry. If so, describe the reflections and/or rotations that map the figure onto itself. ANSWER: line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = 0, the reflection in the line y = x, and the reflection in the line y = -x all map the square onto itself; the rotations of 90, 180, and 270 degrees around the origin map the square onto itself.

27. SOLUTION: This triangle is scalene, so it cannot have symmetry.

ANSWER: no symmetry 29. SOLUTION: The trapezoid has line symmetry, because it is isosceles, but it does not have rotational symmetry, because no trapezoid does.

The reflection in the line y = 1.5 maps the trapezoid onto itself, because that is the perpendicular bisector 28. to the parallel sides. SOLUTION: ANSWER: This figure is a square, because each pair of adjacent sides is congruent and perpendicular. line symmetry; the reflection in the line y = 1.5 maps All squares have both line and rotational symmetry. the trapezoid onto itself. The line symmetry is vertically, horizontally, and diagonally through the center of the square, with lines that are either parallel to the sides of the square or that include two vertices of the square. The equations of those lines are: x = 0, y = 0, y = x, and y = -x

The rotational symmetry is for each quarter turn in a 30. square, so the rotations of 90, 180, and 270 degrees around the origin map the square onto itself. SOLUTION: This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its ANSWER: center, which is the point (1, -1.5). line symmetry; rotational symmetry; the reflection in the line x = 0, the reflection in the line y = 0, the Since this parallelogram is not a rhombus it does not reflection in the line y = x, and the reflection in the have line symmetry. line y = -x all map the square onto itself; the rotations of 90, 180, and 270 degrees around the origin map ANSWER: the square onto itself. rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram onto itself.

SOLUTION: a. Describe the two-dimensional symmetry created The trapezoid has line symmetry, because it is by the photo. isosceles, but it does not have rotational symmetry, b. Is there rotational symmetry in the photo? Explain because no trapezoid does. your reasoning.

The reflection in the line y = 1.5 maps the trapezoid SOLUTION: onto itself, because that is the perpendicular bisector a Sample answer: There is a horizontal line of to the parallel sides. symmetry between the tower and its reflection. There is a vertical line of symmetry through the ANSWER: center of the photo. line symmetry; the reflection in the line y = 1.5 maps b No; sample answer: Because of how the image is the trapezoid onto itself. reflected over the horizontal line, there is no rotational symmetry.

ANSWER: a. Sample answer: There is a horizontal line of symmetry between the tower and its reflection. There is a vertical line of symmetry through the center of the photo. 30. b No; sample answer: Because of how the image is reflected over the horizontal line, there is no SOLUTION: rotational symmetry. This figure is a parallelogram, so it has rotational symmetry of a half turn or 180 degrees around its COORDINATE GEOMETRY Determine center, which is the point (1, -1.5). whether the figure with the given vertices has line symmetry and/or rotational symmetry. Since this parallelogram is not a rhombus it does not 32. R(–3, 3), S(–3, –3), T(3, 3) have line symmetry. SOLUTION: ANSWER: Draw the figure on a coordinate plane. rotational symmetry; the rotation of 180 degrees around the point (1, -1.5) maps the parallelogram onto itself.

A figure has line symmetry if the figure can be a. Describe the two-dimensional symmetry created mapped onto itself by a reflection in a line.

by the photo. b. Is there rotational symmetry in the photo? Explain The given triangle has a line of symmetry through your reasoning. points (0, 0) and (–3, 3). SOLUTION: a Sample answer: There is a horizontal line of A figure in the plane has rotational symmetry if the symmetry between the tower and its reflection. figure can be mapped onto itself by a rotation There is a vertical line of symmetry through the between 0° and 360° about the center of the figure. center of the photo. There is not way to rotate the figure and have it map b No; sample answer: Because of how the image is onto itself. reflected over the horizontal line, there is no rotational symmetry. Thus, the figure has only line symmetry.

ANSWER: ANSWER: a. Sample answer: There is a horizontal line of line symmetry between the tower and its reflection. There is a vertical line of symmetry through the 33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) center of the photo. b No; sample answer: Because of how the image is SOLUTION: reflected over the horizontal line, there is no Draw the figure on a coordinate plane. rotational symmetry.

A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The given figure has 4 lines of symmetry. The line of symmetry are though the following pairs of points {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, and {(2, 2), (2, –2)}.

A figure in the plane has rotational symmetry if the

figure can be mapped onto itself by a rotation A figure has line symmetry if the figure can be between 0° and 360° about the center of the figure.

mapped onto itself by a reflection in a line. The figure can be rotated from the origin and map onto itself. The order of symmetry is 4. The given triangle has a line of symmetry through points (0, 0) and (–3, 3). Thus, the figure has both line symmetry and rotational symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation ANSWER: between 0° and 360° about the center of the figure. line and rotational There is not way to rotate the figure and have it map onto itself. 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 2) Thus, the figure has only symmetry. line SOLUTION: ANSWER: Draw the figure on a coordinate plane. line

33. A(–4, 0), B(0, 4), C(4, 0), D(0, –4) SOLUTION: Draw the figure on a coordinate plane.

A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The given hexagon has 2 lines of symmetry. The lines pass through the following pair of points {(0, 4), (0, – 4)}, and {(3, 0), (–3, 0)}

A figure in the plane has rotational symmetry if the A figure has line symmetry if the figure can be figure can be mapped onto itself by a rotation mapped onto itself by a reflection in a line. The between 0° and 360° about the center of the figure. given figure has 4 lines of symmetry. The line of The figure has rotational symmetry. You can rotate symmetry are though the following pairs of points the figure once within 360° and have it map to itself. {(4, 0), (–4, 0)}, {(0, 4) (0, –4)}, {(–2, 2), (–2, –2)}, and {(2, 2), (2, –2)}. Thus, the figure has both line symmetry and rotational symmetry. A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation ANSWER: between 0° and 360° about the center of the figure. line and rotational The figure can be rotated from the origin and map onto itself. The order of symmetry is 4. 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3)

SOLUTION: Thus, the figure has both line symmetry and Draw the figure on a coordinate plane. rotational symmetry.

ANSWER: line and rotational 34. F(0, –4), G(–3, –2), H(–3, 2), J(0, 4), K(3, 2), L(3, – 2) SOLUTION:

Draw the figure on a coordinate plane. A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. The trapezoid has a line of reflection through points (0,3) and (0, –3).

A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation between 0° and 360° about the center of the A figure has line symmetry if the figure can be figure. There is no way to rotate this figure and have mapped onto itself by a reflection in a line. The it map onto itself. Thus, it does not have rotational given hexagon has 2 lines of symmetry. The lines symmetry. pass through the following pair of points {(0, 4), (0, – 4)}, and {(3, 0), (–3, 0)} Therefore, the figure has only line symmetry.

ANSWER: A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation line between 0° and 360° about the center of the figure. ALGEBRA The figure has rotational symmetry. You can rotate Graph the function and determine the figure once within 360° and have it map to itself. whether the graph has line and/or rotational symmetry. If so, state the order and magnitude of symmetry, and write the equations of any lines of Thus, the figure has both line symmetry and symmetry. rotational symmetry. 36. y = x ANSWER: SOLUTION: line and rotational Graph the function. 35. W(–2, 3), X(–3, –3), Y(3, –3), Z(2, 3) SOLUTION: Draw the figure on a coordinate plane.

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The

line y = x has reflectional symmetry since any line A figure has line symmetry if the figure can be perpendicular to y = x is a line of reflection. The mapped onto itself by a reflection in a line. The equation of the line symmetry is y = –x. trapezoid has a line of reflection through points (0,3) and (0, –3). A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation A figure in the plane has rotational symmetry if the between 0° and 360° about the center of the figure. figure can be mapped onto itself by a rotation The line can be rotated twice within 360° and be between 0° and 360° about the center of the mapped onto itself. figure. There is no way to rotate this figure and have it map onto itself. Thus, it does not have rotational The number of times a figure maps onto itself as it symmetry. rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational Therefore, the figure has only line symmetry. symmetry.

ANSWER: The magnitude of symmetry is the smallest angle line through which a figure can be rotated so that it maps onto itself. ALGEBRA Graph the function and determine The graph has magnitude of symmetry of whether the graph has line and/or rotational . symmetry. If so, state the order and magnitude of symmetry, and write the equations of any lines of Thus, the graph has both reflectional and rotational symmetry. symmetry. 36. y = x SOLUTION: ANSWER: Graph the function. rotational; 2; 180°; line symmetry; y = –x

2 A figure has reflectional symmetry if the figure can 37. y = x + 1 be mapped onto itself by a reflection in a line. The line y = x has reflectional symmetry since any line SOLUTION: perpendicular to y = x is a line of reflection. The Graph the function. equation of the line symmetry is y = –x.

The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of symmetry. The graph has order 2 rotational A figure has reflectional symmetry if the figure can symmetry. be mapped onto itself by a reflection in a line. The graph is reflected through the y-axis. Thus, the The magnitude of symmetry is the smallest angle equation of the line symmetry is x = 0. through which a figure can be rotated so that it maps onto itself. A figure in the plane has rotational symmetry if the The graph has magnitude of symmetry of figure can be mapped onto itself by a rotation . between 0° and 360° about the center of the figure. There is no way to rotate the graph and have Thus, the graph has both reflectional and rotational it map onto itself. symmetry. Thus, the graph has only reflectional symmetry. ANSWER: rotational; 2; 180°; line symmetry; y = –x ANSWER: line; x = 0

3-5 Symmetry

2 3 37. y = x + 1 38. y = –x SOLUTION: SOLUTION: Graph the function. Graph the function.

A figure has reflectional symmetry if the figure can A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The be mapped onto itself by a reflection in a line. The graph is reflected through the y-axis. Thus, the graph does not have a line of reflections where the equation of the line symmetry is x = 0. graph can be mapped onto itself.

A figure in the plane has rotational symmetry if the A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation figure can be mapped onto itself by a rotation between 0° and 360° about the center of the between 0° and 360° about the center of the figure. There is no way to rotate the graph and have figure. You can rotate the graph through the origin it map onto itself. and have it map onto itself.

Thus, the graph has only reflectional symmetry. The number of times a figure maps onto itself as it rotates from 0° to 360° is called the order of ANSWER: symmetry. The graph has order 2 rotational line; x = 0 symmetry.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps onto itself. The graph has magnitude of symmetry of .

Thus, the graph has only rotational symmetry.

ANSWER: rotational; 2; 180° 38. y = –x3 SOLUTION: Graph the function.

39. Refer to the rectangle on the coordinate plane. eSolutions Manual - Powered by Cognero Page 16

A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. The graph does not have a line of reflections where the graph can be mapped onto itself.

A figure in the plane has rotational symmetry if the a. What are the equations of the lines of symmetry of figure can be mapped onto itself by a rotation the rectangle? between 0° and 360° about the center of the b. What happens to the equations of the lines of figure. You can rotate the graph through the origin and have it map onto itself. symmetry when the rectangle is rotated 90 degrees counterclockwise around its center of symmetry? The number of times a figure maps onto itself as it Explain. rotates from 0° to 360° is called the order of SOLUTION: symmetry. The graph has order 2 rotational a. The lines of symmetry are parallel to the sides of symmetry. the rectangles, and through the center of rotation.

The magnitude of symmetry is the smallest angle through which a figure can be rotated so that it maps The slopes of the sides of the rectangle are 0.5 and onto itself. The graph has magnitude of symmetry of -2, so the slopes of the lines of symmetry are the . same. The center of the rectangle is (1, 1.5). Use the Thus, the graph has only rotational symmetry. point-slope formula to find equations.

ANSWER:

rotational; 2; 180° b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

ANSWER: a. b. The equations of the lines of symmetry do not

change; although the rectangle does not map onto 39. Refer to the rectangle on the coordinate plane. itself under this rotation, the lines of symmetry are mapped to each other.

40. MULTIPLE REPRESENTATIONS In this problem, you will use dynamic geometric software to investigate line and rotational symmetry in regular polygons.

a. Geometric Use The Geometer’s Sketchpad to a. What are the equations of the lines of symmetry of draw an equilateral triangle. Use the reflection tool the rectangle? under the transformation menu to investigate and b. What happens to the equations of the lines of determine all possible lines of symmetry. Then record symmetry when the rectangle is rotated 90 degrees their number. counterclockwise around its center of symmetry? b. Geometric Use the rotation tool under the transformation menu to investigate the rotational Explain. symmetry of the figure in part a. Then record its SOLUTION: order of symmetry. a. The lines of symmetry are parallel to the sides of c. Tabular Repeat the process in parts a and b for a the rectangles, and through the center of rotation. square, regular pentagon, and regular hexagon. Record the number of lines of symmetry and the

order of symmetry for each polygon. The slopes of the sides of the rectangle are 0.5 and d. Verbal Make a conjecture about the number of -2, so the slopes of the lines of symmetry are the lines of symmetry and the order of symmetry for a same. regular polygon with n sides. The center of the rectangle is (1, 1.5). Use the point-slope formula to find equations. SOLUTION: a. Construct an equilateral triangle and label the vertices A, B, and C. Draw a line through A perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image b. The equations of the lines of symmetry do not maps to the original, then this line is a line of change; although the rectangle does not map onto reflection. itself under this rotation, the lines of symmetry are mapped to each other.

ANSWER: a. b. The equations of the lines of symmetry do not change; although the rectangle does not map onto itself under this rotation, the lines of symmetry are mapped to each other.

MULTIPLE REPRESENTATIONS 40. In this Next, draw a line through B perpendicular to . problem, you will use dynamic geometric software to Reflect the triangle in the line. Show the labels of the investigate line and rotational symmetry in regular reflected image. If the image maps to the original, polygons. then this line is a line of reflection.

a. Geometric Use The Geometer’s Sketchpad to draw an equilateral triangle. Use the reflection tool under the transformation menu to investigate and determine all possible lines of symmetry. Then record their number. b. Geometric Use the rotation tool under the transformation menu to investigate the rotational symmetry of the figure in part a. Then record its order of symmetry. c. Tabular Repeat the process in parts a and b for a Lastly, draw a line through C perpendicular to . square, regular pentagon, and regular hexagon. Reflect the triangle in the line. Show the labels of the Record the number of lines of symmetry and the reflected image. If the image maps to the original, order of symmetry for each polygon. then this line is a line of reflection. d. Verbal Make a conjecture about the number of lines of symmetry and the order of symmetry for a regular polygon with n sides. SOLUTION: a. Construct an equilateral triangle and label the vertices A, B, and C. Draw a line through A perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

There are 3 lines of symmetry.

b. Construct an equilateral triangle and show the labels of the vertices. Next, find the center of the triangle. Since this is an equilateral triangle, the Next, draw a line through B perpendicular to . circumcenter, incenter, centroid, and orthocenter are Reflect the triangle in the line. Show the labels of the the same point. Construct altitudes through each reflected image. If the image maps to the original, vertex and label the intersection. then this line is a line of reflection. Rotate the triangle about point D. A 120 degree rotation will map the image to the original. Show the labels of the image.

Lastly, draw a line through C perpendicular to . Reflect the triangle in the line. Show the labels of the reflected image. If the image maps to the original, then this line is a line of reflection.

Rotate the triangle again about point D. A 240 degree rotation will map the image to the original. Show the labels of the image

There are 3 lines of symmetry. The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original. b. Construct an equilateral triangle and show the labels of the vertices. Next, find the center of the triangle. Since this is an equilateral triangle, the circumcenter, incenter, centroid, and orthocenter are the same point. Construct altitudes through each vertex and label the intersection. Rotate the triangle about point D. A 120 degree rotation will map the image to the original. Show the labels of the image.

Rotate the triangle again about point D. A 240 the original when reflected in each of the 4 lines degree rotation will map the image to the original. constructed. So there are 4 lines of symmetry. Show the labels of the image Next, rotate the square about the center point. The image maps to the original at 90, 180, 270, and 360 degree rotations. So the order of symmetry is 4.

The triangle can be rotated a third time about D. A 360 degree rotation maps the image to the original.

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and 360 degree rotations. So the order of symmetry is 5.

degree rotations. So the order of symmetry is 4.

Regular Pentagon Construct a regular pentagon and then construct lines through each vertex perpendicular to the sides. Use the reflection tool first to find that the image maps onto the original when reflected in each of the 5 lines constructed. So there are 5 lines of symmetry. Next, rotate the square about the center point. The image maps to the original at 72, 144, 216, 288, and

360 degree rotations. So the order of symmetry is 5.

ANSWER: a. 3 b. 3 c.

Regular Hexagon Construct a regular hexagon and then construct lines through each vertex perpendicular to the sides. Use d. Sample answer: A regular polygon with n sides the reflection tool first to find that the image maps has n lines of symmetry and order of symmetry n. onto the original when reflected in each of the 6 lines ERROR ANALYSIS constructed. So there are 6 lines of symmetry. 41. Jaime says that Figure A has Next, rotate the square about the center point. The only has line symmetry, and Jewel says that Figure A image maps to the original at 60, 120, 180, 240, 300, has only rotational symmetry. Is either of them

and 360 degree rotations. So the order of symmetry correct? Explain your reasoning. is 6.

SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. This figure has 4 lines of symmetry.

A figure in the plane has rotational symmetry if the d. Sample answer: for each figure studied, the figure can be mapped onto itself by a rotation number of sides of the figure is the same as the lines between 0° and 360° about the center of the figure. of symmetry and the order of symmetry. A regular polygon with n sides has n lines of symmetry and order of symmetry n.

ANSWER: a. 3 b. 3

c. The figure also has rotational symmetry.

Therefore, neither of them are correct. Figure A has both line and rotational symmetry.

d. Sample answer: A regular polygon with n sides ANSWER: has n lines of symmetry and order of symmetry n. Neither; Figure A has both line and rotational symmetry. 41. ERROR ANALYSIS Jaime says that Figure A has only has line symmetry, and Jewel says that Figure A 42. CHALLENGE A quadrilateral in the coordinate has only rotational symmetry. Is either of them plane has exactly two lines of symmetry, y = x – 1 correct? Explain your reasoning. and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of symmetry. SOLUTION: Graph the figure and the lines of symmetry.

SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. This figure has 4 lines of symmetry.

Pick points that are the same distance a from one line and the same distance b from the other line. In the same answer, the quadrilateral is a rectangle with sides which are parallel to the lines of symmetry. A figure in the plane has rotational symmetry if the This guarantees that the vertices of the quadrilateral figure can be mapped onto itself by a rotation are the same distance a from one line and the same between 0° and 360° about the center of the figure. distance b from the other line. In this case, a = and b = .

A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2).

The figure also has rotational symmetry. ANSWER: Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) Therefore, neither of them are correct. Figure A has both line and rotational symmetry.

ANSWER: Neither; Figure A has both line and rotational symmetry.

42. CHALLENGE A quadrilateral in the coordinate plane has exactly two lines of symmetry, y = x – 1 and y = –x + 2. Find a set of possible vertices for the figure. Graph the figure and the lines of 43. REASONING A figure has infinitely many lines of symmetry. symmetry. What is the figure? Explain. SOLUTION: SOLUTION: Graph the figure and the lines of symmetry. circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines.

ANSWER: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines.

44. OPEN-ENDED Draw a figure with line symmetry

Pick points that are the same distance a from one but not rotational symmetry. Explain. line and the same distance b from the other line. In SOLUTION: the same answer, the quadrilateral is a rectangle with A figure has line symmetry if the figure can be sides which are parallel to the lines of symmetry. mapped onto itself by a reflection in a line. A figure This guarantees that the vertices of the quadrilateral in the plane has rotational symmetry if the figure can are the same distance a from one line and the same be mapped onto itself by a rotation between 0° and distance b from the other line. In this case, a = 360° about the center of the figure.

and b = . Identify a figure that has line symmetry but does not have rotational symmetry. A set of possible vertices for the figure are, (–1, 0), (2, 3), (4, 1), and (1, 2). An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does ANSWER: not have rotational symmetry because it cannot be Sample answer: (–1, 0), (2, 3), (4, 1), and (1, –2) rotated from 0° to 360° and map onto itself.

ANSWER: Sample answer: An isosceles triangle has line symmetry from the vertex angle to the base of the triangle, but it does not have rotational symmetry 43. REASONING A figure has infinitely many lines of because it cannot be rotated from 0° to 360° and symmetry. What is the figure? Explain. map onto itself. SOLUTION: circle; Every line through the center of a circle is a line of symmetry, and there are infinitely many such lines. ANSWER: 45. WRITING IN MATH How are line symmetry and circle; Every line through the center of a circle is a rotational symmetry related? line of symmetry, and there are infinitely many such lines. SOLUTION: In both types of symmetries the figure is mapped 44. OPEN-ENDED Draw a figure with line symmetry onto itself. but not rotational symmetry. Explain. Rotational symmetry. SOLUTION: A figure has line symmetry if the figure can be mapped onto itself by a reflection in a line. A figure in the plane has rotational symmetry if the figure can Reflectional symmetry: be mapped onto itself by a rotation between 0° and 360° about the center of the figure.

Identify a figure that has line symmetry but does not have rotational symmetry.

An isosceles triangle has line symmetry from the In some cases an object can have both rotational and vertex angle to the base of the triangle, but it does reflectional symmetry, such as the diamond, however not have rotational symmetry because it cannot be some objects do not have both such as the crab. rotated from 0° to 360° and map onto itself.

ANSWER: Sample answer: An isosceles triangle has line symmetry from the vertex angle to the base of the ANSWER: triangle, but it does not have rotational symmetry Sample answer: In both rotational and line symmetry because it cannot be rotated from 0° to 360° and a figure is mapped onto itself. However, in line map onto itself. symmetry the figure is mapped onto itself by a reflection, and in rotational symmetry, a figure is mapped onto itself by a rotation. A figure can have line symmetry and rotational symmetry.

46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of symmetry and the order of symmetry, and then she 45. WRITING IN MATH How are line symmetry and enters this value into a database. Which value should rotational symmetry related? she enter in the database for the tile shown here? SOLUTION: In both types of symmetries the figure is mapped onto itself.

Rotational symmetry.

A Reflectional symmetry: 2 B 3 C 4 D 8 SOLUTION:

In some cases an object can have both rotational and reflectional symmetry, such as the diamond, however some objects do not have both such as the crab. The tile is a rhombus and has 2 lines of symmetry. Each connects opposite corners of the tile.

It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn around its center.

2 + 2 = 4, so C is the correct answer.

ANSWER: ANSWER: Sample answer: In both rotational and line symmetry C a figure is mapped onto itself. However, in line symmetry the figure is mapped onto itself by a 47. Patrick drew a figure that has rotational symmetry reflection, and in rotational symmetry, a figure is but not line symmetry. Which of the following could mapped onto itself by a rotation. A figure can have be the figure that Patrick drew? line symmetry and rotational symmetry. A 46. Sasha owns a tile store. For each tile in her store, she calculates the sum of the number of lines of

symmetry and the order of symmetry, and then she B enters this value into a database. Which value should she enter in the database for the tile shown here?

A 2

B 3 E C 4 D 8 SOLUTION: SOLUTION: Option A has rotational and reflectional symmetry.

The tile is a rhombus and has 2 lines of symmetry. Option B has reflectional symmetry but not rotational Each connects opposite corners of the tile. symmetry.

It has an order of symmetry of 2, because it has rotational symmetry at 180 degrees, or each half turn around its center.

2 + 2 = 4, so C is the correct answer. Option C has neither rotational nor reflectional ANSWER: symmetry. C

47. Patrick drew a figure that has rotational symmetry but not line symmetry. Which of the following could be the figure that Patrick drew? Option D has rotational symmetry but not reflectional A symmetry.

Option E has reflectional symmetry but not rotational symmetry. C

E The correct choice is D.

ANSWER: SOLUTION: D Option A has rotational and reflectional symmetry. 48. Which of the following figures may have exactly one line of symmetry and no rotational symmetry? A Equilateral triangle B Equiangular triangle C Isosceles triangle Option B has reflectional symmetry but not rotational D Scalene triangle symmetry. SOLUTION: An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

ANSWER: Option C has neither rotational nor reflectional C symmetry. 49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry? Option D has rotational symmetry but not reflectional symmetry. A B C D

SOLUTION: Option E has reflectional symmetry but not rotational First, plot the points. symmetry.

The correct choice is D.

ANSWER: D

48. Which of the following figures may have exactly one Then, plot each option A-D to consider each figure line of symmetry and no rotational symmetry? and its symmetry. A Equilateral triangle Option A has both reflectional and rotational B Equiangular triangle symmetry. C Isosceles triangle D Scalene triangle SOLUTION: An isosceles triangle has one line of symmetry and no rotational symmetry. The correct choice is C.

ANSWER: C

49. Camryn plotted the points , and . Which of the following additional points can she plot so that the resulting quadrilateral PQRS has line symmetry but not rotational symmetry? Option B has reflective symmetry but not rotational symmetry. The correct choice is B. A B C D SOLUTION: First, plot the points.

Then, plot each option A-D to consider each figure and its symmetry. Option A has both reflectional and rotational symmetry.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

In order for the figure to map onto itself, the line of reflection must go through the center point.

Two lines of reflection go through the sides of the figure.

Two lines of reflection go through the vertices of the figure.

Thus, there are four possible lines that go through the center and are lines of reflections.

Therefore, the figure has four lines of symmetry.

State whether the figure appears to have line ANSWER: symmetry. Write yes or no. If so, copy the yes; 4 figure, draw all lines of symmetry, and state their number.

1. SOLUTION: A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line.

The figure has reflectional symmetry.

2. In order for the figure to map onto itself, the line of SOLUTION: reflection must go through the center point. A figure has reflectional symmetry if the figure can be mapped onto itself by a reflection in a line. Two lines of reflection go through the sides of the figure. The given figure does not have reflectional symmetry. There is no way to fold or reflect it onto itself.