Concurrent Lines, Medians,

and Altitudes Section 5.3 Vocab Word: Concurrent When three or more lines intersect in one point,

they are "concurrent."

The point at which they intersect is the "point of concurrency." This figure shows triangle UTV with the bisectors of its angles concurrent at I. THe point of concurrency of the angle bisectors of a triangle is called the "incenter of the triangle." . Theorem 5.6 The perpendicular bisectors of the sides of a

triangle are concurrent at a point equidistant

from the vertices Theorem 5.7 The bisectors of the angles of a triangle are

concurrent at a point equidistant from the sides Vocabulary Word: Circumcenter The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle. Points Q, R, and S are equidistant from C, the circumcenter. The circle is "circumscribed about" the triangle. Example: For use with Mini Whiteboards!! Find the center of the circle that you can circumscribe about the triangle with vertices (0,0), (-8,0), and (0,6). Points X, Y, and Z are equidistant from I, the incenter. The circle is insribed in the triangle. Vocabulary Word: Median A "median of a triangle" is a segment whose endpoints are a vertex and the midpoint of the opposite side. Theorem 5.8 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. In a triangle, the point of concurrency of the medians is the "Centroid." The point is also called the center of gravity because it is the point where a triangular shape will balance. In triangle ABC, D is the centroid and DE=6. Find BE. Altitude: All Attitude An "altitude of a triangle" is the perpendicular segment from a vertex to the line containing the opposite side. Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle or it may lie outside the triangle. The lines containing the altitudes of a triangle are concurrent at the "orthocenter of the triangle." Theorem 5.9 The lines that contain the altitudes of a triangle are concurrent. Identifying Medians and Altitudes (a) Is ST a median, an altitude, or neither?

(b) Is UW a median, an altitude, or neither? Worksheet

Homework: Pg. 275 (1, 8-16, 19-22, 28)

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# HG 5.3: Concurrent Lines, Medians, and Altitudes

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