1. 5.2 Bisectors of a Triangle Goal: To use segment bisectors and perpendicular lines to solve problems involving triangles and real world scenarios.

2. Perpendicular Bisector of a Triangle A perpendicular bisector of a triangle is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side. Perpendicular Bisector

3. A point is equidistant from two figures if the point is the same distance from each figure. Theorem 5.2 Perpendicular Bisector Theorem In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

5. Theorem 5.3 Converse of the Perpendicular Bisector Theorem: In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Ex.1: BD is the perpendicular bisector of AC. Find AD.

6. Ex.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?

7. Ex.3: In the diagram, JK is the perpendicular bisector of NL. a. What segment lengths are equal? Explain your reasoning. b. Find NK. c. Explain why M is on JK?

8. Ex.4: KM is the perpendicular bisector of JL. Find JK and ML.

9. Concurrency When three or more lines, rays, or segments intersect in the same point, they are called concurrent lines, rays, or segments. The point of intersection of the lines, rays, or segments is called the point of concurrency.

10. About concurrency The three perpendicular bisectors of a triangle are concurrent. The point of concurrency may be inside the triangle, on the triangle, or outside the triangle. 90° Angle- Right Triangle

11. About concurrency The three perpendicular bisectors of a triangle are concurrent. The point of concurrency may be inside the triangle, on the triangle, or outside the triangle. Acute Angle- Acute Scalene Triangle

12. About concurrency The three perpendicular bisectors of a triangle are concurrent. The point of concurrency may be inside the triangle, on the triangle, or outside the triangle. Obtuse Angle- Obtuse Scalene Triangle

13. Theorem 5.4 Concurrency of Perpendicular Bisectors of a Triangle: The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. The point of concurrency (point of intersection) of the three perpendicular bisectors of a triangle is called the circumcenter of the triangle.

14. FACILITIES PLANNING. A company plans to build a distribution center that is convenient to three of its major clients. The planners start by roughly locating the three clients on a sketch and finding the circumcenter of the triangle formed.

15. Practice 2x = 5x – 6 0 = 3x – 6 6 = 3x x = 2 AB = 4 3x + 8 = 7x – 16 8 = 4x – 16 24 = 4x x = 6 AB = 26 6x + 11 = 11x – 9 11 = 5x – 9 20 = 5x x = 4 AB = 35