5.7 Reflections and Symmetry

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Goal A reflection is a transformation that creates a mirror image. The Identify and use original figure is reflected in a line that is called the line of reflection . reflections and lines of symmetry. PROPERTIES OF REFLECTIONS Key Words r • image p. 152 ●1 The reflected image is congruent to the original figure. • reflection • line of symmetry ●2 The orientation of the reflected image is reversed. ●3 The line of reflection is the perpendicular bisector of line of the segments joining the original reflection image corresponding points.

EXAMPLE 1 Identify Reflections Tell whether the red triangle G is the reflection of the blue triangle in line m. F H m

FЈ HЈ

Visualize It! Solution GЈ clockwise orientation Check to see if all three properties of a reflection are met. G ●1 Is the image congruent to the original figure? Yes. ✔ ●2 Is the orientation of the image reversed? Yes. ✔ F H TFGH has a clockwise orientation. FЈ HЈ TFЈGЈHЈ has a counterclockwise orientation.

GЈ ●3 Is m the perpendicular bisector of the segments connecting the corresponding points? Yes. ✔ G counterclockwise orientation F H To check, draw a m diagram and connect the corresponding FЈ HЈ endpoints.

GЈ

ANSWER ᮣ Because all three properties are met, the red triangle is the reflection of the blue triangle in line m.

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EXAMPLE 2 Identify Reflections Tell whether the red triangle is the m reflection of the blue triangle in line m.

Solution Check to see if all three properties of a reflection are met. ●1 Is the image congruent to the original figure? Yes. ✔ ●2 Is the orientation of the image reversed? No. ANSWER ᮣ The red triangle is not a reflection of the blue triangle.

Student Help EXAMPLE 3 Reflections in a Coordinate Plane

VOCABULARY TIP a. Which segment is the reflection B(Ϫ1, 3) y E(1, 3) Use the following of AB&* in the x-axis? Which point relationship to help you remember that a corresponds to A? to B? 1 A(Ϫ4, 1) D(4, 1) reflection is a flip: b. Which segment is the reflection J(Ϫ4, Ϫ1) 1 x re fl ection of AB&* in the y-axis? Which point fl ip corresponds to A? to B? K(Ϫ1, Ϫ3) Solution a. The x-axis is the perpendicular bisector of AJ&* and BK&* , so the reflection of AB&* in the x-axis is JK&* . A(Ϫ4, 1) → J(Ϫ4, Ϫ1) A is reflected onto J. B(Ϫ1, 3) → K(Ϫ1, Ϫ3) B is reflected onto K.

b. The y-axis is the perpendicular bisector of AD&* and BE&* , so the reflection of AB&* in the y-axis is DE&* . A(Ϫ4, 1) → D(4, 1) A is reflected onto D. B(Ϫ1, 3) → E(1, 3) B is reflected onto E.

Identify Reflections

Tell whether the red figure is a reflection of the blue figure. If the red figure is a reflection, name the line of reflection.

1.y 2.y 3. y 1

1 1 1 x

1 x 1 x

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Symmetry In the photo, the mirror’s edge creates a line of symmetry. A figure in the plane has a line of symmetry if the figure can be reflected onto itself by a reflection in the line.

A line of symmetry is a line of reflection.

Visualize It! EXAMPLE 4 Determine Lines of Symmetry Determine the number of lines of symmetry in a square. Solution Think about how many different ways you can fold a square so that the edges of the figure match up perfectly. You may want to draw a shape on paper, cut vertical fold horizontal fold diagonal fold diagonal fold it out, and then fold it to find the lines of symmetry.

ANSWER ᮣ A square has four lines of symmetry.

EXAMPLE 5 Determine Lines of Symmetry Determine the number of lines of symmetry in each figure. a. b. c.

Solution a. 2 lines of symmetry b. no lines of symmetry c. 6 lines of symmetry

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EXAMPLE 6 Use Lines of Symmetry Kaleidoscopes Mirrors are used to create images seen through a kaleidoscope. The angle between the mirrors is aA. eyepiece

black glass mirror casing mirror

mirror mirror angle A glass colored glass or Top view Image seen by viewer cover liquid

KALEIDOSCOPES The parts Find the angle measure used to create the kaleidoscope design. Use of a kaleidoscope are shown 18 0Њ the equation maA ϭ ᎏᎏ , where n is the number of lines of above. n Application Links symmetry in the pattern. CLASSZONE.COM a. b. c.

Solution a. The design has 3 lines of symmetry. So, in the formula, n ϭ 3. 18 0Њ 18 0Њ maA ϭ ᎏᎏ ϭ ᎏᎏ ϭ 60 Њ n 3 b. The design has 4 lines of symmetry. So, in the formula, n ϭ 4. 18 0Њ 18 0Њ maA ϭ ᎏᎏ ϭ ᎏᎏ ϭ 45 Њ n 4 c. The design has 6 lines of symmetry. So, in the formula, n ϭ 6. 18 0Њ 18 0Њ maA ϭ ᎏᎏ ϭ ᎏᎏ ϭ 30 Њ n 6

Determine Lines of Symmetry

Determine the number of lines of symmetry in the figure.

4. 5. 6.

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5.75.7 Exercises Guided Practice

Vocabulary Check 1. Complete the statement: A figure in the plane has a(n) __?__ if the figure can be reflected onto itself by a(n) __?__ in the line.

Skill Check Determine whether the red figure is a reflection of the blue figure.

2. 3. 4. m

Flowers Determine the number of lines of symmetry in the flower. 5. 6. 7.

Practice and Applications

Extra Practice Identifying Reflections Determine whether the figure in red is a See p. 684. reflection of the figure in blue. Explain why or why not. 8. 9. 10. m m m

Reflections in a Coordinate Plane Tell whether the grid shows a reflection in the x-axis , the y-axis , or neither .

11. 12. 13. Homework Help y C y B C y B D B D Example 1: Exs. 8–10 1 1 Example 2: Exs. 8–10 D A Example 3: Exs. 11–16 A 3 x E 1 x Example 4: Exs. 21–29 E H 1 Example 5: Exs. 21–29 A C Example 6: Exs. 37–39 1 x F F G

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Student Help Reflections in a Coordinate Plane In Exercises 14–16, use the diagram at the right. SKILLS REVIEW y To review coordinates, 14. Which segment is the reflection of AB&* in the D see p. 664. x-axis? Which point corresponds to A? to B? B &* A C 15. Which segment is the reflection of AB in the 1 y-axis? Which point corresponds to A? to B? 1 x 16. Compare the coordinates for AB&* with the G E coordinates for its reflection in the x-axis. How are the coordinates alike? How are H F they different?

Visualize It! Trace the figure and draw its reflection in line k. 17. 18. 19. k k

20. Paper Folding Follow these steps. ●1 Fold a piece of paper in half, twice. ●2 Draw a triangle and cut it out. B C ●3 Unfold the paper and label the A D sections. Which of the triangles are reflections of the triangle in section A? Explain.

Symmetry Decide whether the line shown is a line of symmetry. 21. 22. 23.

Lines of Symmetry Determine the number of lines of symmetry. 24. 25. 26.

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You be the Judge Determine whether all lines of symmetry are shown. If not, sketch the figure and draw all the lines of symmetry. 27. 28. 29.

30. Visualize It! A piece of paper is folded in half and some cuts are made as shown. Sketch the figure that represents the piece of paper unfolded. Careers Type Design In Exercises 31 and 32, use the lowercase letters of the alphabet shown below.

31. Which letters are reflections of other letters? 32. Draw each letter that has at least one line of symmetry and sketch its line(s) of symmetry. Which letters have one line of symmetry? Which letters have two lines of symmetry? TYPE DESIGNERS design fonts that appear in books, magazines, newspapers, and Word Reflections Determine if the entire word has any lines of other materials that we read symmetry. If so, write the word and draw the line(s) of symmetry. every day. Erik Spiekermann, shown above, has designed 33. 34. 35. 36. many fonts that are widely used today. Career Links CLASSZONE.COM

Kaleidoscope Designs Find the measure of the angle between the mirrors ( aA) that produces the kaleidoscope design. Use the equation 0؇ 18 .ᎏᎏ ؍ maA n 37. 38. 39.

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EXAMPLE Show Triangles are Congruent Show that TABC c T JKL . y B (5, 4) Solution A(1, 2) C (5, 2) Show that the corresponding sides 1 are congruent. 1 x For sides on a horizontal grid line, J (1, Ϫ2) L(5, Ϫ2) subtract the x-coordinates. CA ϭ 5 Ϫ 1 ϭ 4 LJ ϭ 5 Ϫ 1 ϭ 4 K (5, Ϫ4) For sides on a vertical grid line, subtract the y-coordinates. BC ϭ 4 Ϫ 2 ϭ 2 KL ϭ ؊4 Ϫ (؊2) ϭ Ϫ2 ϭ 2 Student Help For any other sides, use the distance formula. LOOK BACK ϭ Ϫ 2 ϩ Ϫ 2 ϭ 2 ϩ 2 ϭ For help with the AB ͙ෆ(5ෆෆ1ෆ) ෆෆ(ෆ4ෆෆ2ෆ) ͙ෆ4ෆෆෆ2ෆ ͙2ෆ0ෆ distance formula, JK ϭ Ϫ 2 ϩ ؊ Ϫ ؊ 2 ϭ 2 ϩ Ϫ 2 ϭ see p. 194. ͙ෆ(5ෆෆ1ෆ) ෆෆ(ෆ( ෆ4ෆෆ(ෆෆ2ෆ)) ෆ ͙ෆ4ෆෆෆ(ෆෆ2ෆ) ͙2ෆ0ෆ By the SSS Congruence Postulate, TABC c T JKL .

Showing Triangles are Congruent In Exercises 40 and 41, refer to the example above. Show that TABC c T DEF . 40. 41. y y C(3, 4) B(2, 3)

1 1 A(2, 1) C(6, 1) B(1, 1) A(3, 1) 1 D(2, Ϫ1) F (6, Ϫ1) x E(Ϫ1, Ϫ1) 1 x

D(Ϫ1, Ϫ3) E(2, Ϫ3) F (Ϫ4, Ϫ3)

Standardized Test 42. Multiple Choice Which triangle shows y J Practice the image when TXYZ is reflected in L the y-axis? D E 1 ɕA TDEF ɕB TJKL F K Z 1 R x ɕC TPQR ɕD None of these X Y Œ P

43. Multiple Choice How many lines of symmetry does the figure at the right have?

ɕF 0 ɕG 1

ɕH 2 ɕJ 3

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Mixed Review Showing Lines are Parallel Find the value of x so that p ʈ q. (Lesson 3.5)

44. 45. 46. p q ؇ p ؇ p 92 ؇ 82 105 q x ؉ 10) ؇) x؇ q (3 x ؊ 1) ؇

Finding Angle Measures Find the measure of a1. (Lesson 4.2) 47. 48. 49. ؇ 44 ؇ 51 1

1 ؇ 75 ؇ 1 30 ؇ 38

Algebra Skills Comparing Numbers Compare the two numbers. Write the answer (Skills Review, p. 662) .؍ using >, <, or

50. 2348 and 2384 51. Ϫ5 and Ϫ7 52. 19.1 and 19.01

53. Ϫ11.2 and Ϫ11.238 54. 0.065 and 0.056 55. 1.011 and 1.11

Quiz 3

1. Sketch the overlapping triangles separately. A D Mark all congruent angles and sides. Which postulate or theorem can you use to show that the triangles are congruent? B C (Lesson 5.5)

Use the diagram to find the indicated measure(s). (Lesson 5.6) 2. Find DC . 3. Find ML and JK . 4. Find AB .

L B A M 3x ؉ 4 2 x ؉ 7 2 4 9 25 D J A D C B 4 C K

Determine the number of lines of symmetry in the figure. (Lesson 5.7)

5. 6. 7.

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