Perpendiculars and Bisectors Lesson 7.2 Perpendiculars and Bisectors pp. 267-272

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.

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

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.

Circumcenter Z W V P c U X Y a b

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.

Incenter D

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.

altitude of a triangle

x x median of a triangle

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

Orthocenter B P A C

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

Centroid Q P R

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

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

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

Homework pp. 270-272

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

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

■ 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?

■ 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?

■ 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?

■ 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?

■ 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?