1. Section 5 – 3 Concurrent Lines, Medians, and Altitudes
Objectives:
To identify properties of perpendicular bisectors and angle bisectors
To identify properties of medians and altitudes of triangles

2. Concurrent: Point of Concurrency:
When three or more lines intersect in one point.
Point of Concurrency:
The point at which concurrent lines intersect.

3. Theorem 5 - 6
The perpendicular bisector of the sides of a triangle are concurrent at a point equidistant from the vertices.
Circumcenter of the Triangle:
The point of concurrency of the perpendicular bisectors of a triangle.

4. Theorem 5 - 7
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.
Incenter of the Triangle:
The point of concurrency of the angle bisectors of a triangle.

5. Point Q, R, and S are equidistant from the circumcenter, so the circle is circumscribed about the triangle.
Point X, Y, and Z are equidistant from the incenter, so the circle is inscribed in the triangle.

6. Example 1 Finding the Circumcenter
A) Find the center of the circle that you can circumscribe about ∆OPS.

7. B) Find the center of the circle that you can circumscribe about the triangle with vertices (0,0), (-8, 0), and (0, 6).

8. C) Find the center of the circle that circumscribes ∆XYZ.

9. Example 2 Real-World Connection
A) The Jacksons want to install the largest possible circular pool in their triangular backyard. Where would the largest possible pool be located?

10. Textbook Page 259 – 260; #1 – 9
(USE GRAPH PAPER)

11. Median of a Triangle:
A segment whose endpoints are a vertex and the midpoint of the opposite side

12. Theorem 5 - 8
The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.
Centroid of the Triangle:
The point of concurrency of the medians.

13. Example 3 Finding Lengths of Medians
A) D is the centroid of ∆ABC and DE = 6. Find BD. Then find BE.

14. B) M is the centroid of ∆WOR, and WM = 16. Find WX.

15. Altitude of a Triangle:
The perpendicular segment from a vertex to the line containing the opposite side.

16. Example 4 Identifying Medians & Altitudes
A) B)

17. Theorem 5 - 9
The lines that contain the altitudes of a triangle are concurrent.
Orthocenter of the Triangle:
The point of concurrency of the altitudes.

18. Textbook Page 260; # 11 – 22