Find each measure of MN. Justify Perpendicular Bisector Theorem

Write an equation to solve for a. Justify 3a + 20 = 2a + 26 Converse of  Bisector Theorem

Find the measures of BD and BC. Justify BD = 12 BC =24 Converse of  Bisector Theorem

Find the measure of BC. Justify BC = 7.2  Bisector Theorem

Write the equation to solve for x. Justify your equation. 3x + 9 = 7x – 17  Bisector Theorem

Find the measure. mEFH, given that mEFG = 50°. Justify m EFH = 25 Converse of the  Bisector Theorem

Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints C(6, –5) and D(10, 1).

Perpendicular Bisectors of a triangle… 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. C

Angle Bisectors Paste-able! of a triangle… 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. I

DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC.

MZ and LZ are perpendicular bisectors of ∆GHJ. Find GM

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.

5-3: Medians and Altitudes Z Y X C B  Medians of triangles: Endpoints are a vertex 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. P

5-3: Medians and Altitudes 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.

In ∆LMN, RL = 21 and SQ =4. Find LS.

In ∆LMN, RL = 21 and SQ =4. Find NQ.

In ∆JKL, ZW = 7, and LX = 8.1. Find KW.

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).