Chapter 5.2 Notes: Use Perpendicular Bisectors Goal: You will use perpendicular Bisectors.

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Presentation transcript:

Chapter 5.2 Notes: Use Perpendicular Bisectors Goal: You will use perpendicular Bisectors.

A segment, ray, line or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. 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.

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.

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?

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?

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

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.

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.

Ex.5: The perpendicular bisectors of ΔABC meet at point G. Find GB. Ex.6: The perpendicular bisectors of ΔRST meet at point D. Find DR and DU.

Ex.7: Three snack carts sell frozen yogurt from points A, B, and C outside a city. Each of the three carts is the same distance from the frozen yogurt distributor. Find a location for the distributor that is equidistant from the three carts.

Circumcenter – Acute Triangle: Circumcenter is inside the triangle. – Right Triangle: Circumcenter is on the triangle. – Obtuse Triangle: Circumcenter is outside the triangle.