Basic Geometric Concepts
Basic Geometric Concepts
The Lesson Activities will help you meet these educational goals:
· Content Knowledge—You will know precise definitions of angle, circle, perpendicular line, parallel line, and line segment.
· Mathematical Practices—You will make sense of problems and solve them, use appropriate tools strategically, look for and make use of structure, and look for and express regularity in repeated reasoning.
· Inquiry—You will perform an investigation in which you will make observations, analyze results, and communicate your results in written form.
· 21st Century Skills—You will employ online tools for research and analysis, apply creativity and innovation, use critical-thinking and problem-solving skills, communicate effectively, and carry out technology-assisted modeling.
Directions
You will evaluate some of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.
______
Self-Checked Activities
Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.
1. Line Segments and Angles
In this activity, you will use the GeoGebra geometry tool to explore the properties of line segments and angles. Open GeoGebra, and then complete each step below. If you need help, follow these instructions for using GeoGebra.
a. Plot three points on the coordinate plane and label them A, B, and C. (Be sure that all three points do not lie in a straight line.) Now join the points two at a time using straight paths. How many unique straight paths can you make through the points? Which geometric figure is formed?
Sample answer:
It is possible to make three unique straight paths, and Together, the three paths form a triangle.
b. The three paths intersect in pairs at the points A, B, and C. Measure and record the angle formed at each intersection. Also measure and record the lengths of the straight paths, or sides. Capture the figure showing the three angles and three sides, and paste it in the space below.
Sample answer:
Answers will vary depending on the points chosen. This sample answer is for the points A(3, 4), B(1, 1), and C(9, 1).
m = 56.31°
m = 97.13°
m = 26.57°
AB = 3.61 units
BC = 8 units
AC = 6.71 units
c. The three straight paths, and meet each other at three points, A, B, and C. How do these points of intersection differ from each other? Explain the differences in terms of the angles that you see. Also look at the length of the side opposite each angle. What pattern do you see regarding the measurements? In what situation would all the points of intersection resemble one another? Modify the triangle in GeoGebra to help you with your answers.
Sample answer:
The three straight paths intersect at different angles. In other words, the amount required to turn one path onto the other is different about each vertex. The largest angle is opposite the longest side, and the smallest angle is opposite the shortest side. If all three points of intersection look the same, the result is a triangle with three equal angles and three equal sides.
d. Use GeoGebra to move points A, B, and C to the locations shown by the ordered pairs in the table. Record the length of each side and the measure of each angle for the resulting triangle. Note that the length of a line segment is denoted with two letters but no bar on top. For example, AB is read as “the length of ”
Sample answer:
A(3, 4), B(1, 1), C(5, 1) / 3.61 units / 4 units / 3.61 units / 56.31° / 67.38° / 56.31°
A(4, 5), B(2, 1), C(7, 3) / 4.47 units / 5.39 units / 3.61 units / 41.63° / 82.87° / 55.49°
A(3, 6), B(3, -2), C(-3, -2) / 8 units / 6 units / 10 units / 90° / 36.87° / 53.13°
e. If you move point C toward point B along how does m change? How do and change? What happens to these angles if you move point C away from point B along
Sample answer:
If point C moves toward point B along m does not change. However, m decreases and m increases. As point C moves away from point B, the opposite happens: mincreases, mdecreases, but m stays the same.
f. Delete and focus on the intersection of and Keeping points B and C fixed, move point A to different locations and observe how changes. What must you do to increase or decrease m? What happens if you move point A to a location directly on ? What is m at this location? Explain.
Sample answer:
If the direction of A with respect to B changes, mchanges. The angle increases or decreases depending on the location of point A. This assumes is fixed. The measure of equals 0° when A becomes a point on In other words, no motion is necessary to turn onto in this case.
2. Circles
In this activity, you will use the GeoGebra geometry tool to explore the properties of circles. Open GeoGebra, and then complete each step below.
a. Plot the point A(5, 5). Using a line tool, create with a length of 4 units from point A. Turn on the trace feature at point B, and move point B around point A, keeping the length of fixed. Capture the image, and paste it in the space below.
Sample answer:
b. Create a circle such that its center is point A and B is a point on the circle. Capture the image, and paste it in the space below. How does the shape you traced in this part of the activity compare with the circle you created in part a?
Sample answer:
The shape traced in part a is the same as the circle created in part b.
c. In parts a and b, what is the distance from the center of the circle to a point on the circle? Take this idea a step further: what is the relationship between the center of any circle and the points that lie on the circle?
Sample answer:
The distance from any point on this circle to the center is 4 units—the distance between A and B. For any circle, the distance between the center and any point on the circle is fixed.
3. Perpendicular Lines
Use GeoGebra to explore the properties of perpendicular lines. Then complete each step below.
a. Move point B to different locations in the coordinate plane. What do you notice about the relationship between and ? Explain in terms of
Sample answer:
is always perpendicular to The measure of is always 90°.
b. Move point B some more. As you move point B, the angle formed betweenand varies. If you want to makeperpendicular to what do you need to do? Explain in terms of
Sample answer:
To make perpendicular to move point B until the angle between the lines, measures 90°.
c. Describe the relationship betweenandwhen the two lines are perpendicular to Also describe the relationship betweenandwhenis not perpendicular to Zoom in or out on the coordinate plane, if needed.
Sample answer:
When and are perpendicular to they do not intersect each other. When is not perpendicular to and intersect at a single point.
4. Parallel Lines
Use GeoGebra to explore the properties of parallel lines. Then complete each step below.
a. Study the ordered pairs in the table. Move point D to the locations shown, and note m, m, AE, and BF. You can turn on the grid to find the locations more easily. Round angle measurements to the nearest degree.
Sample answer:
(12, 2) / 85° / 95° / 2.72 units / 3.29 units
(11, 1) / 81° / 99° / 2.03 units / 3.04 units
(4, 2) / 90° / 90° / 3.62 units / 3.62 units
(8, 5) / 102° / 78° / 5.70 units / 4.37 units
(9, 7) / 109° / 71° / 6.93 units / 4.82 units
b. Based on your observations from part a, what is the relationship between and when and measure something other than 90°? In this situation, what is the relationship between and ? Explain.
Sample answer:
If and measure something other than 90°, then ≠ In this case, and intersect at a single point.
c. What condition is necessary for and to be equal? In this situation, what is the relationship between and ? What can you conclude about parallel lines with regard to the distance between them? Use the word perpendicular in your answer.
Sample answer:
In order for to equal m and m must both be 90°. When this occurs, does not intersect the two lines are parallel. and represent the perpendicular distances between and When the two lines are parallel, the perpendicular distance between them is the same throughout their length.
2