5-1 Bisectors, Medians, and Altitudes 1.) Identify and use perpendicular bisectors and angle bisectors in triangles. 2.) Identify and use medians and altitudes in triangles.

5-1 Bisectors, Medians, and Altitudes perpendicular bisector - a line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side. Theorem 5.1 Perpendicular Bisector Theorem Theorem 5.2 Converse of Perpendicular Bisector Theorem

5-1 Bisectors, Medians, and Altitudes The perpendicular bisector does not have to start from a vertex! Example: In the scalene ∆CDE, is the perpendicular bisector. A B In the right ∆MLN, is the perpendicular bisector. In the isosceles ∆POQ, is the perpendicular bisector. Since a triangle has three sides, how many perpendicular bisectors do triangles have?? Perpendicular bisectors of a triangle intersect at a common point.

5-1 Bisectors, Medians, and Altitudes Perpendicular bisectors of a triangle intersect at a common point. Concurrent Lines: Three or more lines that intersect at a common point. Point of Concurrency: Point of intersection for concurrent lines. Circumcenter: Point of concurrency of the perpendicular bisectors of an triangle. **The circumcenter does not have to belong inside the triangle. Circumcenter

5-1 Bisectors, Medians, and Altitudes Theorem 5.3 Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle. **If J is the circumcenter of ABC, then AJ = BJ = CJ. A B C J

Example 1: BD is the perpendicular bisector of AC. Find AD 5-1 Bisectors, Medians, and Altitudes

Example 2: In the diagram, WX is the perpendicular bisector of YZ. (a) What segment lengths in the diagram are equal? (b) Is V on WX? 5-1 Bisectors, Medians, and Altitudes

Example 3: In the diagram, JK is the perpendicular bisector of NL. (a) Find NK. (b) Explain why M is on JK. 5-1 Bisectors, Medians, and Altitudes

Example 4: In the diagram, BC is the perpendicular bisector of AD. Find the value of x. 5-1 Bisectors, Medians, and Altitudes

Theorem 5.4 Angle Bisector Theorem Theorem 5.5 Converse of Angle Bisector Theorem

5-1 Bisectors, Medians, and Altitudes Theorem 5.6 Incenter Theorem The incenter of a triangle is equidistant from each side of the triangle. **If G is the incenter of ABC, then GE = GD = GF. AF is angle bisector of <BAC BD is angle bisector of <ABC CE is angle bisector of <BCA Angle bisectors of a triangle intersect at a common point called the incenter. Angle bisectors of a triangle are congruent.

5-1 Bisectors, Medians, and Altitudes Example 5: Find the measure of angle GFJ if FJ bisects <GFH.

5-1 Bisectors, Medians, and Altitudes Example 6: Find the value of x.

5-1 Bisectors, Medians, and Altitudes Example 7: Find the value of x.

5-1 Bisectors, Medians, and Altitudes Example 8: QS is the angle bisector of <PQR. Find the value of x.

5-1 Bisectors, Medians, and Altitudes Classwork: Study Guide and Intervention p. 55 Extra problems: p. 242 #6, p. 243 #16

5-1 Bisectors, Medians, and Altitudes Median -- a segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex. Since there are three vertices, there are three medians. In the figure C, E and F are the midpoints of the sides of the triangle. The medians of a triangle also intersect at a common point called the centroid.

5-1 Bisectors, Medians, and Altitudes Theorem 5.7 Centroid Theorem The centroid of a triangle is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. **If L is the centroid of ABC, then AL = AE, BL = BF CL = CD L

5-1 Bisectors, Medians, and Altitudes Example 1: Points U, V, and W are the midpoints of YZ, ZX, and XY. Find a, b, and c. X W Y U Z V 7.4 2a c 3b + 2

5-1 Bisectors, Medians, and Altitudes Example 2: Points T, H, and G are the midpoints of MN, MK, and NK. Find w, x, and y. M T K H N G 2x y 3.2 3w - 2

5-1 Bisectors, Medians, and Altitudes Altitude -- a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. Every triangle contains 3 altitudes. The altitudes of a triangle also intersect at a common point called the orthocenter. Altitudes of an acute triangle. Altitudes of an obtuse triangle.