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5-1 Perpendicular and Angle Bisectors Holt Geometry 5-1 Perpendicular and Angle Bisectors Holt Geometry 5-1 Perpendicular and Angle Bisectors Find each measure of MN. Justify MN = 2.6 Perpendicular Bisector Theorem Holt Geometry 5-1 Perpendicular and Angle Bisectors Write an equation to solve for a. Justify 3a + 20 = 2a + 26 Converse of Bisector Theorem Holt Geometry 5-1 Perpendicular and Angle Bisectors Find the measures of BD and BC. Justify BD = 12 BC =24 Converse of Bisector Theorem Holt Geometry 5-1 Perpendicular and Angle Bisectors Find the measure of BC. Justify BC = 7.2 Bisector Theorem Holt Geometry 5-1 Perpendicular and Angle Bisectors Write the equation to solve for x. Justify your equation. 3x + 9 = 7x – 17 Bisector Theorem Holt Geometry 5-1 Perpendicular and Angle Bisectors Find the measure. mEFH, given that mEFG = 50°. Justify m EFH = 25 Converse of the Bisector Theorem Holt Geometry 5-1 Perpendicular and Angle Bisectors Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints C(6, –5) and D(10, 1). Holt Geometry Perpendicular Bisectors of a triangle… C • bisect each side at a right angle • meet at a point called the circumcenter • The circumcenter is equidistant from the 3 vertices of the triangle. • The circumcenter is the center of the circle that is circumscribed about the triangle. • The circumcenter could be located inside, outside, or ON the triangle. Angle Bisectors of a triangle… I • bisect each angle • meet at the incenter • The incenter is equidistant from the 3 sides of the triangle. • The incenter is the center of the circle that is inscribed in the triangle. • The incenter is always inside the circle. DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC. GC = 13.4 MZ and LZ are perpendicular bisectors of ∆GHJ. Find GM GM = 14.5 Z is the circumcenter of ∆GHJ. GK and JZ GK = 18.6 JZ = 19.9 Find the circumcenter of ∆HJK with vertices H(0, 0), J(10, 0), and K(0, 6). MP and LP are angle bisectors of ∆LMN. Find the distance from P to MN. MP and LP are angle bisectors of ∆LMN. Find mPMN. mPMN = 30 5-3: Medians and Altitudes and Angle Bisectors B 5-1 Perpendicular Medians of triangles: X P Y •Endpoints are a vertex A C Z and midpoint of opposite side. •Intersect at a point called the centroid •Its coordinates are the average of the 3 vertices. •The centroid is ⅔ of the distance from each vertex to the midpoint of the opposite 2 2 2 AP AY BP BZ CP CX side. 3 3 3 •The centroid is always located inside the triangle. Holt Geometry 5-3: Medians and Altitudes and Angle Bisectors 5-1 Perpendicular Altitudes of a triangle: • A perpendicular segment from a vertex to the line containing the opposite side. • Intersect at a point called the orthocenter. • An altitude can be inside, outside, or on the triangle. Holt Geometry In ∆LMN, RL = 21 and SQ =4. Find LS. LS = 14 In ∆LMN, RL = 21 and SQ =4. Find NQ. 12 = NQ In ∆JKL, ZW = 7, and LX = 8.1. Find KW. KW = 21 Example 2: Problem-Solving Application A sculptor is shaping a triangular piece of iron that will balance on the point of a cone. At what coordinates will the triangular region balance? Find the average of the x-coordinates and the average of the y-coordinates of the vertices of ∆PQR. Make a conjecture about the centroid of a triangle. Find the orthocenter of ∆XYZ with vertices X(3, –2), Y(3, 6), and Z(7, 1). X