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Geometry Theorems and Theorems: Explained and Applied

Explore the fundamental theorems of geometry, including Circumcenter, Incenter, Centroid, and Orthocenter. Learn about the properties of perpendicular bisectors and angle bisectors in triangles. Find practical examples applying these theorems in algebraic contexts.

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Geometry Theorems and Theorems: Explained and Applied

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  1. NM Standards: AFG.C.5, GT.A.5, GT.B.4, DAP.B.3

  2. Any point that is on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. M B N B M P D If is an angle bisector, then PM = PN P A C N D A C If PM = PN, then is an angle bisector. Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. • Any point on the angle bisector is equidistant from the sides of the angle. • Any point equidistant from the sides of an angle lies on the angle bisector. Remember, distance is always measured on the perpendicular.

  3. Definitions Point of Concurrency - A common point in which three or more lines intersect Circumcenter - The intersection point of the three perpendicular bisectors of a triangle Incenter - The intersection point of the three angle bisectors of a triangle Centroid - The intersection point of the three medians of a triangle Orthocenter - The intersection point of the three altitudes of a triangle

  4. Theorems Circumcenter Theorem- The circumcenter of a triangle is equidistant from the vertices of the triangle Incenter Theorem- The incenter of a triangle is equidistant from each side of the triangle Centroid Theorem- The centroid of a triangle is two-thirds the distance of its corresponding median of the triangle

  5. Example 1-2a ALGEBRA Points U, V, and W are the midpoints of respectively. Find a,b, and c. Find a. Segment Addition Postulate Centroid Theorem Substitution Multiply each side by 3 and simplify. Subtract 14.8 from each side. Divide each side by 4.

  6. Example 1-2a Find b. Segment Add Postulate Centroid Theorem Find c. Segment Addition Postulate Centroid Theorem

  7. Example 1-2b ALGEBRA Points T, H, and G are the midpoints of respectively. Find w,x, and y. Answer:

  8. HW : Page 242

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