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 3 a + 20 = 2 a + 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. 3 x + 9 = 7 x – 17 Bisector Theorem Holt Geometry

5 -1 Perpendicular and Angle Bisectors Find the measure. m EFH, given that m EFG = 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.

Paste-able! 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 m PMN = 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 side. • 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