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Section 5.2 U se Perpendicular Bisectors

Section 5.2 U se Perpendicular Bisectors. Vocabulary P erpendicular Bisector : A segment, ray, line, or plane that is perpendicular to a segment at its midpoint E quidistant : A point that is the same distance from each figure.

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Section 5.2 U se Perpendicular Bisectors

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  1. Section 5.2 Use Perpendicular Bisectors

  2. Vocabulary • Perpendicular Bisector: A segment, ray, line, or plane that is perpendicular to a segment at its midpoint • Equidistant: A point that is the same distance from each figure. • Concurrent: When three or more lines, rays, or segments intersect at the same point

  3. Point of Concurrency: The point of intersection of concurrent lines, rays, or segments • Circumcenter: The point of concurrency of the three perpendicular bisectors of a triangle

  4. 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. C A B P If CP is the bisector of AB, then CA = CB.

  5. Theorem 5.3: Converse of the Perpendicular Bisector Theorem C In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. A B P D If DA = DB, then D lies on the bisector of AB.

  6. Example 1 Use the Perpendicular Bisector Theorem AC is the perpendicular bisector of BD. Find AD. B 7x-6 A C AD = AB 4x = 7x - 6 x = 2 AD = 4x = 4(2) = 8 4x D

  7. Example 2 Use perpendicular bisectors K N J L 13 13 M In the diagram, KN is the perpendicular bisector of JL. a. What segment lengths in the diagram are equal? b. Is M on KN? a. KN bisects JL, so NJ = NL. Because K is on the perpendicular bisector of JL, KJ = KL by Thm 5.2. The diagram shows MJ = ML = 13.

  8. K N J L 13 13 M b. Because MJ = ML, M is equidistant from J and L. So, by the Converse of the Perpendicular Bisector Theorem, M is on the perpendicular of JL, which is KN.

  9. With a partner, do 1-2 on p. 265

  10. checkpoints! F 4.1 4.1 J K H G 2x x+1 In the diagram, JK is the perpendicular bisector of GH. 1. What segment lengths are equal? 2. Find GH.

  11. 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 PF are perpendicular bisectors, then PA = PB = PC. E D P A C F

  12. Example 3 Use the concurrency of perpendicular bisectors N Q P S 9 2 O M R The perpendicular bisectors of ΔMNO meet at point S. Find SN. Using Theorem 5.4, you know that point S is equidistant from the vertices of the triangle. So, SM = SN = SO. SN = SM SN = 9

  13. Do #3 on p. 265

  14. B 12 E D G 15 6 C A F checkpoint! The perpendicular bisectors of ΔABC meet at point G. Find GC.

  15. Essential Question Whatdo you know about perpendicular bisectors?

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