1. 5.1 Bisector, Medians, and Altitudes

4. B. Theorem 5.1: If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.
Example 1: Find x, if AD is a perpendicular bisector.
6x - 5
3x + 16 A B C D

5. C. Theorem 5.3: Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle.
Example 2
Find x and y, if BE = 24,
AE = 3x – 6 and CE = 5y + 4 A B C E

6. D. Theorem 5.4: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.
Example 3
Find x, if AC is an angle bisector of  DAB and DC = x + 4 and BC = 2x – 5.  D C A B

7. E. Theorem 5.7: Centroid Theorem The distance from the vertex to the point of concurrency is equal to 2 times the distance from the point of concurrency to the midpoint of the opposite side.
Example 4
Point A is a centroid of ∆DEF. Find x, y, and z. D S E T F U A
SA = 4z
EA = y
TA = 2x – 5
FA = 4.6
UA = 2.9
DA = 6