Lesson 5.3.notebook November 28, 2012 Warm‐up Use the figure below for problems 1 and 2. 1. Find BD 2. Find CA Use the figure below for problem 3. 3. Find the perimeter of quadrilateral ABCG 1 Lesson 5.3.notebook November 28, 2012 Page 267 Homework Answers 2 Lesson 5.3.notebook November 28, 2012 Proofs Page 267 3 Lesson 5.3.notebook November 28, 2012 Proofs Page 267 4 Lesson 5.3.notebook November 28, 2012 Lesson 5.3 ‐ Concurrent Lines, Medians, and Altudes Vocabulary Concurrent lines are three or more lines that intersect in one point. The point of concurrency is the point at which concurrent lines intersect. A circle is circumscribed about a polygon when the verces of the polygon are on the circle The circumcenter of a triangle is the point of concurrency of the perpendicular bisectors of a triangle. A median of a triangle is a segment whose endpoints are on a vertex and the midpoint of the opposite side. 5 Lesson 5.3.notebook November 28, 2012 Vocabulary A circle is inscribed in a polygon if the sides of the polygon are equidistant from the incenter of the polygon. The incenter of a triangle is the point of concurrency of the angle bisectors of a triangle. The altude of a triangle is the perpendicular segment from a vertex to a line containing the opposite side. The centroid of a triangle is the point of concurrency of the medians. The orthocenter of a triangle is the point of intersecon of the lines containing the altudes of the triangles. 6 Lesson 5.3.notebook November 28, 2012 Vocabulary and Key Concepts Theorem 5.6 The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the verces. Theorem 5.7 The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. 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. Theorem 5.9 The lines that contain the altudes of a triangle are concurrent. 7 Lesson 5.3.notebook November 28, 2012 Ex 1: Find the center of the circle that circumscribes ΔXYZ 8 Lesson 5.3.notebook November 28, 2012 Ex 2: M is the centroid of ΔWOR, and WM = 16. Find WX. 9 Lesson 5.3.notebook November 28, 2012 Ex 3: In ΔABC, centriod D is on median AM. AD = x + 4 and DM = 2x ‐ 6. Find AM. 10 Lesson 5.3.notebook November 28, 2012 Homework: Page 275 #1‐9, 11‐16, 19‐22, 27‐30, 37‐39 11.