Bisectors Medians and Altitudes Skill 27 Objective HSGC

Bisectors, Medians, and Altitudes Skill 27

Objective HSG-C. 3/10: Students are responsible for using properties of bisectors, medians and altitudes. Also, using these properties to understand problems.

Definitions When three or more lines intersect at one point they are called, concurrent. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of a triangle. Using the circumcenter as the center of a circle the circle will contain each vertex and the circle is circumscribed about the triangle.

Definitions The point of concurrency of the angle bisectors of a triangle is called the incenter of a triangle. A circle with center at the incenter of a triangle is inscribed in the triangle. A median of a triangle is a segment whose endpoints are a vertex and the midpoint of the opposite side.

Definitions The point of concurrency of the medians of a triangle is the centroid of a triangle. An altitude of a triangle is the perpendicular segment from a vertex of the triangle to the line containing the opposite side. The point of concurrency of the altitudes of a triangle is the orthocenter of a triangle.

Theorem 28: Concurrency of Perp. Bisectors Thm. The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

Theorem 29: Concurrency of Angle Bisectors Thm. The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.

Theorem 30: Concurrency of Medians Theorem 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.

Theorem 31: Concurrency of Altitudes Theorem The lines that contain the altitudes of a triangle are concurrent.

Example 1; Using the Circumcenter A town planer wants to locate a new fire station equidistant from the elementary, middle, and high schools. Where should the station be located? Explain. Need to find the circumcenter, b/c it is equidistant from each vertex. E H F M The fire station should be placed at point F.

Example 2; Identifying and Using the Incenter G is the incenter b/c all of the angle bisectors meet at G. GD, GE, and GF are the distance from G to each side, by definition of distance A D G C F E B

Example 3; Identifying the Length of the Median The distance from a vertex to the centroid of a triangle is 2/3 the total length. Y B C A X Z

Example 4; Identifying Medians and Altitudes P T R S Q Altitude Median

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