1. Perpendiculars and Bisectors
Lesson 7.2
Perpendiculars and Bisectors pp

2. Objectives:
1. To identify and prove the essential properties of perpendicular bisectors.
2. To state the relationship between specific lines associated with triangles.
3. To identify the special points of concurrency in a triangle.

3. Theorem 7.5
Any point lies on the perpendicular bisector of a segment if and only if it is equidistant from the two endpoints. C A B D

4. Theorem 7.6
Circumcenter Theorem. The perpendicular bisectors of the sides of any triangle are concurrent at the circumcenter, which is equidistant from each vertex of the triangle.

5. Circumcenter Z W V P c U X Y a b

6. Theorem 7.7
Incenter Theorem. The angle bisectors of the angles of a triangle are concurrent at the incenter, which is equidistant from the sides of the triangle.

7. Incenter D

8. Definition
An altitude of a triangle is a segment that extends from a vertex and is perpendicular to the opposite side.
A median of a triangle is a segment extending from a vertex to the midpoint of the opposite side.

9. altitude of a triangle

10. x x
median of a triangle

11. Theorem 7.8
Orthocenter Theorem. The lines that contain the three altitudes are concurrent at the orthocenter.

12. Orthocenter B P A C

13. Theorem 7.9
Centroid Theorem. The three medians of a triangle are concurrent at the centroid.

14. Centroid Q P R

15. The angle bisectors are concurrent at the _____.
1. Orthocenter
2. Centroid
3. Incenter
4. Circumcenter

16. The altitudes are concurrent at the _____.
1. Orthocenter
2. Centroid
3. Incenter
4. Circumcenter

17. Which point of concurrency is illustrated here?
1. Orthocenter
2. Centroid
3. Incenter
4. Circumcenter

18. Homework pp

19. 1. Label the circumcenter C.
►A. Exercises
Draw four obtuse triangles. Use one triangle for each of the next four exercises.
1. Label the circumcenter C. C

20. 3. Label the orthocenter O.
►A. Exercises
Draw four obtuse triangles. Use one triangle for each of the next four exercises.
3. Label the orthocenter O. O

21. ■ Cumulative Review
Given noncollinear points A, B, and C, consider the following:
AB, , AB, AB, AB, AB
22. Which symbol above is not a set?

22. ■ Cumulative Review
Given noncollinear points A, B, and C, consider the following:
AB, , AB, AB, AB, AB
23. Which set listed above is not a subset of any of the other sets?

23. ■ Cumulative Review
Given noncollinear points A, B, and C, consider the following:
AB, , AB, AB, AB, AB
24. Which set is a subset of all of the sets?

24. ■ Cumulative Review
Given noncollinear points A, B, and C, consider the following:
AB, , AB, AB, AB, AB
25. AB is a subset of which other sets?

25. ■ Cumulative Review
Given noncollinear points A, B, and C, consider the following:
AB, , AB, AB, AB, AB
26. For which two of the sets is neither a subset of the other?