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Geometry Theorems and Examples in Triangles

Learn about key theorems such as Bisector, Medians, and Altitudes in triangles with examples to find unknown values. Explore properties related to perpendicular bisectors, circumcenters, and centroids.

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Geometry Theorems and Examples in Triangles

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  1. 5.1 Bisector, Medians, and Altitudes

  2. A 3x + 16 6x - 5 B C D 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.

  3. B E C A C. Theorem 5.3: Circumcenter TheoremThe 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

  4. 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

  5. E T S A D U F E. Theorem 5.7: Centroid TheoremThe 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. SA = 4z EA = y TA = 2x – 5 FA = 4.6 UA = 2.9 DA = 6

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