Congruence. Set 5: Bisectors, Medians, and Altitudes Instruction. Student Activities Overview and Answer Key

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1 Instruction Goal: To provide opportunities for students to develop concepts and skills related to identifying and constructing angle bisectors, perpendicular bisectors, medians, altitudes, incenters, circumcenters, centroids, and orthocenters Common Core Standards Experiment with transformations in the plane. G-CO.1. Prove geometric theorems. G-CO.10. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Prove theorems about triangles. Make geometric constructions. G-CO.12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Student Activities Overview and Answer Key Station 1 Students will be given a ruler and a compass. Students will construct and identify angle bisectors. They will derive the ratio between the two segments created by the angle bisector and the other sides of the triangle. Then they will construct and find the incenter based on the angle bisectors and construct a circle inscribed in the triangle. 50

2 Answers 1. B Instruction A 2. about 2.15 in., about 2.85 in., 3 in., 4 in. 3. BD/DC = AB/AC; proportion 4. Triangle sizes will vary. B C D A 5. No, because only one circle that has a center equidistant from each side of the triangle can be inscribed in the triangle. 51 C

3 Instruction Station 2 Students will be given a compass and a ruler. Students will construct isosceles and obtuse triangles. They will construct the perpendicular bisectors. They will find the circumcenter of the triangle and examine its relationship to each vertex of the triangle. Answers interior 3. Answers will vary, but the distances should be equal; the circumcenter is equidistant from all vertices of the triangle exterior 8. Answers will vary, but the distances should be equal; the circumcenter is equidistant from all vertices of the triangle. 52

4 Station 3 Instruction Students will be given a compass and ruler. Students will construct a triangle. Students will construct the medians and centroid of a triangle. Then they will measure the side lengths of the triangle, medians, and distance from the centroid to each vertex. They will derive the relationship between the centroid, medians, and vertices. Answers Answers will vary. 3. Answers will vary. 4. Answers will vary. 5. The centroid is twice as far from a given vertex than it is from the point of intersection of the median to the opposite side from that vertex. 6. No, the centroid is always on the interior of the triangle. By definition, medians are between each vertex and the midpoint of each side. Therefore, the centroid must always be inside the triangle. Station 4 Students will be given a compass and a ruler. Students will construct altitudes and orthocenters for acute, obtuse, and right triangles. They will analyze the length of altitudes and locations of orthocenters. 53

5 Instruction Answers 1. A 2. Answers will vary. 3. inside the triangle 4. B 5. Answers will vary. 6. outside the triangle 7. C 8. Answers will vary. 9. on the vertex of the 90 angle 54

6 Materials List/Setup Station 1 Station 2 Station 3 Station 4 compass; ruler compass; ruler compass; ruler compass; ruler Instruction 55

7 Discussion Guide Instruction To support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to debrief the station activities. Prompts/Questions 1. What is the name of the intersection of the angle bisectors? 2. What is the name of the intersection of the perpendicular bisectors? 3. What is a median of a triangle? 4. What is the name of the intersection of the medians of a triangle? 5. What is an altitude of a triangle? 6. What is the name of the intersection of the altitudes of a triangle? Think, Pair, Share Have students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class. Suggested Appropriate Responses 1. incenter 2. circumcenter 3. The median of a triangle is a line segment drawn from a vertex to the midpoint of the opposite side. 4. centroid 5. The altitude of a triangle is the perpendicular distance from a vertex of a figure to the side opposite the vertex. 6. orthocenter Possible Misunderstandings/Mistakes Confusing how to identify angle bisectors versus perpendicular bisectors Confusing how to identify and construct medians versus altitudes Mixing up the terms incenter, circumcenter, centroid, and orthocenter 56

8 Station 1 At this station, you will find a ruler and a compass. Work as a group to answer the questions. 1. In the space below, construct a right triangle ABC with legs equal to 3 inches and 4 inches, and a hypotenuse of 5 inches. Label the right angle as A. Use the compass to bisect the triangle at vertex A. Name the point where the bisector intersects BC as point D. 2. What is the length of BD? What is the length of DC? What is the length of AB? What is the length of AC? 3. Based on your observations in problems 1 2, how do BD and DC relate to AB and AC? Show your work and answer in the space below. Complete this statement: An angle bisector divides the side opposite the bisected angle into two segments that are the same as the other sides of the triangle. continued 57

9 4. In the space below, construct an acute triangle with angles measuring 40, 80, and 60. Bisect each angle. 5. The point where the angle bisectors meet is called the incenter. On the triangle you constructed for problem 4, use the compass to construct a circle, with the incenter as its center point. Can you inscribe another circle within the triangle that is also equidistant from each side of the triangle? Why or why not? 58

10 Station 2 At this station, you will find a compass and a ruler. Work as a group to answer the questions. 1. In the space below, construct an isosceles triangle with angles measuring 70, 70, and 40. Find the midpoint of each side of the triangle. Construct a perpendicular bisector at the midpoint of each side of the triangle. 2. Do the three perpendicular bisectors meet at a point that is on the interior or exterior of your circle? The point of intersection of the perpendicular bisectors is called the circumcenter. 3. What is the distance from the circumcenter to each vertex in the triangle? Based on your measurements, what can you say about the relationship between the circumcenter and each vertex of the triangle? continued 59

11 4. In the space below, construct an obtuse triangle with angles measuring 120, 35, and Find the midpoint of each side of the triangle. 6. Construct a perpendicular bisector at the midpoint of each side of the triangle. 7. Do the three perpendicular bisectors meet at a point that is on the interior or exterior of your circle? 8. What is the distance from the circumcenter to each vertex in the triangle? Based on your measurements, what can you say about the relationship between the circumcenter and each vertex of the triangle? 60

12 Station 3 At this station, you will find a ruler and a compass. Work as a group to answer the questions. 1. In the space below, construct an acute triangle ABC. Find the midpoint of each side of the triangle. Construct a median by drawing a straight line from each midpoint to the opposite vertex. Label the intersection of the medians as point C. This point is called the centroid of the triangle. 2. What is the length of each side of the triangle you have created? 3. What is the length of each median? 4. What is the length between the centroid and each vertex? continued 61

13 5. Based on your observations in problems 1 4, what can you say about the relationship between the centroid and a given vertex and the centroid and the point of intersection of the median to the opposite side from that vertex? 6. Can the centroid ever occur on the outside of a triangle? Why or why not? 62

14 Station 4 At this station, you will find a ruler and a compass. Work as a group to answer the questions. 1. In the space below, construct an acute triangle. Construct an altitude from each vertex. Label the intersection of the altitudes as point A. This intersection is called the orthocenter. 2. What is the length of each altitude? 3. Is the orthocenter on the inside, on the outside, or on a vertex of the triangle? 4. In the space below, construct an obtuse triangle. Construct an altitude from each vertex. Label the orthocenter as B. continued 63

15 5. What is the length of each altitude? 6. Is the orthocenter on the inside, on the outside, or on a vertex of the triangle? 7. In the space below, construct a right triangle. Construct an altitude from each vertex. Label the orthocenter as C. 8. What is the length of each altitude? 9. Is the orthocenter on the inside, on the outside, or on a vertex of the triangle? 64

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