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# 5.1 Bisector, Medians, and Altitudes

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5.1 Bisector, Medians, and Altitudes

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

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

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

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

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