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Unit 5 Bisectors, Medians, and Altitudes Part 1 Vocab • Concurrent Lines • When three or more lines intersect they are called concurrent lines • Point of Concurrency • The point of intersection for concurrent lines Perpendicular Bisector • A line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side. • Any point on a perpendicular bisector is equidistant from the endpoints of the segment.(theorem 5.1) • Any point that is equidistant from the endpoints of a segment lies on the perpendicular bisector of that segment. (theorem 5.2) • A triangle has three perpendicular bisectors. Circumcenter • The intersection of the three perpendicular bisectors, their point of concurrency. • The circumcenter of a triangle is equidistant from the vertices of a triangle. (Theorem 5.3) B l k Lines j, k, and l are E j This means AG = GC = GB D C G F A bisctors. Also: BE = CE & AF = CF & AD = BD Also: all bisctors form right angles with the segments they bisect. Angle Bisector Theorems • Any point on the angle bisector is equidistant from the sides of the angle. (theorem 5.4) • Any Point equidistant from the sides of an angle lies on the angle bisector. (theorem 5.5) Incenter • The intersection of the three angle bisectors. • The incenter of a triangle is equidistant from each side of the triangle. C j Segments j, k, and l are all angle bisectors. 3 4 B F H This means BF = DF = EF G 5 1 Also: <1 = <2, <3 = <4, <5 = <6 k D A 6 2 E I Also: GH = GI l Median • A segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex. B BDis a median of ABC AD DC C A D Centroid • The point of intersection of the three medians of a triangle • The centroid is the point of balance for any triangle • The centroid is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. (Theorem 5.7) Centroid Example BD, AF,CEis a median of ABC therefore: B 2 1 AG = AF and GH =AF 3 3 F E G 2 C A D 1 BG = BD and GD = BG 3 3 2 1 CG = CE and GE =CE 3 3 // // 2 BP BE 3 2 (10) BE 3 20 BE 3 2 3 2(EN ) AE 2(x) x 17 x 17 1 3 AN AE EN AN x 17 x AN (17) 17 (17) AN 51 Altitude • A segment formed from the vertex to the line containing the opposite side and is perpendicular to the opposite side. • Every triangle has three altitudes, their intersection is know as the orthocenter. Segment BD is an altitude, therefore BDC and BDA are both right angles B C A D Triangle Inequalities Part 2 Exterior Angle Inequality Theorem • If an angle is an exterior angle of a triangle then its measure is greater than the measure of either of its corresponding remote interior angles.(theorem 5.8) 2 m4 > m1 1 3 4 m4 > m2 Triangle Sides and Angles • If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. • If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. Example B C is the largest angle 5 9 B is the middle angle C A is the smallest angle A 7 E DFis the largest side 77° DEis the middle side 65° F D 38° EFis the smallest side Triangle Inequality Theorem • The sum of the lengths of any two sides of a triangle is greater than the length of the third side. • Always check using the two smallest sides, they must be larger than the third. If this is true the numbers will represent a triangle. Example • Do these numbers represent a triangle? 1.) 9, 7, 12 Yes 2.) 5, 5, 10 No 3.) 1, 4, 6 No 4.) 6, 6, 2 Yes