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Concurrent Lines, Medians, Altitudes Point of Concurrency When three or more lines intersect, they are concurrent. All triangles have 4 sets of lines that are concurrent. Angle Bisectors : Point of concurrency is the incenter. Draw congruency marks. Theorem: The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. Angle Bisectors : Point of concurrency is the incenter. Draw congruency marks. A circle can be inscribed in a triangle from the incenter . Theorem: The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. Perpendicular Bisectors: Perpendicular Bisectors: Point of concurrency is the circumcenter. Draw congruency marks. A circle can be circumscribed around a triangle from the circumcenter. Theorem: The perpendicular bisectors of the sides of a triangle are concurrent at a point that is equidistant from the vertices. Median – a segment whose endpoints are a vertex and the midpoint of the opposite side Draw congruency marks. The Centroid is the point of concurrency of medians x 2x 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. If Q is the centroid of triangle ABC and XQ 3 XA QA CZ 6 x 9 y QZ QC a)2 x 9 y b )6 x 9 y c)2 x 3 y d )4 x 6 y Altitude – the perpendicular segment from a vertex to the line containing the opposite side. An altitude can be a side of a triangle, or it can be outside of the triangle. Theorem: The lines that contain the altitudes of a triangle are concurrent. The Orthocenter is the point of concurrency for altitudes The Orthocenter of an obtuse triangle is always outside of the triangle. b. a. e. c. d. Name these lines: b. a. a. b. c. d. Angle bisector Median Altitude Perpendicular bisector e. Midsegment e. c. d. AD BC GF BC BAE CAE BF FC G Markthe theinformation informationgiven. given. Identify Identifythe thefollowing: following: Mark Median - AE Median Altitude - AD Altitude AngleBisector Bisector-AE Angle PerpendicularBisector Bisector FG Perpendicular G Homework: Page276, 277: 8, 11-16 all, 19-22 all, 28 a,b,c. Complete graph QUIZ Monday: Distance of a line segment, midpoint of a line segment, altitude, midsegment, angle bisector, perpendicular bisector, medians. Recall: rise slope run y2 y1 m x2 x1