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Chapter 5.2 Notes: Use Perpendicular Bisectors

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 .

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Chapter 5.2 Notes: Use Perpendicular Bisectors

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

  2. 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.

  3. 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.

  4. 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?

  5. 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?

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

  7. 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.

  8. 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.

  9. 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.

  10. 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.

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

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