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5-2 Use Perpendicular Bisectors

5-2 Use Perpendicular Bisectors. Honors Geometry Ms. Stawicki. Objectives. You will use perpendicular bisectors to solve problems. Perpendicular Bisectors. A segment, ray, line, or plane that is perpendicular to a segment at its midpoint. Equidistant.

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5-2 Use Perpendicular Bisectors

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  1. 5-2Use Perpendicular Bisectors Honors Geometry Ms. Stawicki

  2. Objectives • You will use perpendicular bisectors to solve problems.

  3. Perpendicular Bisectors • A segment, ray, line, or plane that is perpendicular to a segment at its midpoint.

  4. Equidistant • A point is equidistant from two figures if the point is the same distance from each figure. • Points that lie ON the perpendicular bisector of a segment are equidistant from the segment’s endpoints.

  5. Perpendicular Bisector Theorems • Theorem 5-2: Perpendicular Bisector Theorem • If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment C A B P If CP is the perpendicular bisector of AB, then CA = CB.

  6. Perpendicular Bisector Theorems • Theorem 5-3: Converse of the Perpendicular Bisector Theorem • If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment C A B P D If DA = DB, then D lies on the perpendicular bisector of AB.

  7. Example: Use the Perpendicular Bisector Theorem • RS is the perpendicular bisector of PQ. Find PR. P 8x - 9 R S 6x Q

  8. Example: Use Perpendicular Bisectors • JM is the perpendicular bisector of HK. • Which lengths in the diagram are equal? • Is L on JM? H 12.5 J L M 12.5 K

  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 is called the point of concurrency.

  10. Perpendicular Bisector 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. B If PD, PE, and PFare perpendicular Bisectors, then PA= PB = PC. D E P C A F

  11. Circumcenter • The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenter of the triangle. • The circumcenter P is equidistant from the three vertices, so P is the center of a circle that passes through all three vertices. The circle is circumscribed about the triangle. P

  12. Circumcenter • The location of P in relation to the triangle depends on the type of triangle. P P P

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