Line Segments Associated with Triangles Ch. 5-1

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5-1 Special Segments in Triangles

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Line Segments Associated with Triangles Ch. 5-1 Geometry St. Barnabas HS Bronx, NY

Aim: Do Now: Ans: 5 5 miles 4 miles Ans: 3 miles P A B C What is the median, altitude and angle bisector of a triangle? Aim: Do Now: 1) What is the median of the numbers –4, -1, 0, 5, 6, 8, 10 ? Ans: 5 2) What is the altitude in miles of the plane in the diagram? 5 miles 4 miles Ans: 3 miles P A B C Geometry Lesson: Median, Altitude, Angle Bisector

Definition:A median of a triangle is a line segment that joins any vertex of the triangle to the midpoint of the opposite side. Every triangle has three medians. Medians: • • • In each case below, make a congruence statement about 2 line segments. P Q R S T V E F G H L

Definition: An altitude of a triangle is a line segment drawn from any vertex of the triangle, perpendicular to and ending on the opposite side. Every triangle has three altitudes. Altitudes: In each case below, state the name of an altitude and the triangle to which it belongs. 3) A B C 1) B A D C L K P B N G 2)

Defintion: An angle bisector of a triangle is a line segment that bisects any angle of the triangle, and terminates on the side opposite the angle. Every triangle has three angle bisectors. Angle bisectors: ) ) ( P A Q D L X ( Geometry Lesson: Median, Altitude, Angle Bisector

Ex: Special line segments of triangles Q R T X 1) 2) T P X R Q 3) ( T X P R Geometry Lesson: Median, Altitude, Angle Bisector

In ABC, AM is median to BC. If AB = 16, AC = 7, and BC = 11, find BM. BM  MC = 5.5 B BM + MC = BC = 11 B M N C A In the same ABC, BM = 3x – 2 and CM = x + 8. Find the value of x and BC 3x – 2 = x + 8 2x – 2 = 8 2x = 10 x = 5 BC = BM + CM = (3x – 2) + (x + 8) = (3·5 – 2) + (5 + 8) = (13) + (13) = 26

Proofs w/Median, Altitude, Angle Bisector What conclusions can we make based on medians, altitudes and angle bisectors of triangles? Geometry Lesson: Median, Altitude, Angle Bisector

Proofs w/Median, Altitude, Angle Bisector X L 1) Given: Prove: 2) Given: Prove: 3) Given: Prove: Geometry Lesson: Median, Altitude, Angle Bisector

Proofs w/Median, Altitude, Angle Bisector X L • R S 4) Given: Prove: Geometry Lesson: Median, Altitude, Angle Bisector