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Perpendicular & Angle Bisectors

Perpendicular & Angle Bisectors. Objectives. Identify and use ┴ bisectors and  bisectors in ∆s. C. Side AB. A. P. B. perpendicular bisector. Perpendicular Bisector. A ┴ bisector of a ∆ is a line, segment, or ray that passes through the midpoint of one of the sides of

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Perpendicular & Angle Bisectors

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  1. Perpendicular & AngleBisectors

  2. Objectives • Identify and use ┴ bisectors and  bisectors in ∆s

  3. C Side AB A P B perpendicular bisector Perpendicular Bisector A ┴ bisectorof a ∆ is a line, segment, or ray that passes through the midpoint of one of the sides of the ∆ at a 90° .

  4. ┴ Bisector Theorems • Theorem 5.1 – Any point on the ┴ bisector of a segment is equidistant from the endpoints of the segment. • Theorem 5.2 – Any point equidistant from the endpoints of a segment lies on the ┴ bisector of the segment.

  5. C Side AB A P B perpendicular bisector ┴ Bisector Theorems (continued) Basically, if CP is the perpendicular bisector of AB, then PA ≅ PB.

  6. ┴ Bisector Theorems (continued) • Since there are three sides in a ∆, then there are three ┴ Bisectors in a ∆. • These three ┴ bisectors in a ∆ intersect at a common point called the circumcenter.

  7. ┴ Bisector Theorems (continued) • Theorem 5.3 (Circumcenter Theorem)The circumcenter of a ∆ is equidistant from the vertices of the ∆. • Notice, a circumcenter of a ∆ is the center of the circle we would draw if we connected all of the vertices with a circle on the outside (circumscribe the ∆). circumcenter

  8. Example What is the length of AB? A 4x D B 6x-10 C

  9. Angle Bisectors of ∆s • Another special bisector which we have already studied is an  bisector. As we have learned, an  bisector divides an  into two ≅ parts. In a ∆, an  bisector divides one of the ∆s s into two ≅ s.(i.e. if AD is an  bisector then BAD ≅ CAD) B C D

  10. Angle Bisectors of ∆s (continued) • Theorem 5.4 (Angle Bisector Theorem) – Any point on an  bisector is equidistant from the sides of the . • Theorem 5.5 (Converse of the Angle Bisector Theorem) – Any point equidistant from the sides of an  lies on the  bisector.

  11. Angle Bisectors of ∆s (continued) • As with ┴ bisectors, there are three  bisectors in any ∆. These three  bisectors intersect at a common point we call the incenter. incenter

  12. Angle Bisectors of ∆s (continued) • Theorem 5.6 (Incenter Theorem)The incenter of a ∆ is equidistant from each side of the ∆.

  13. Example What is the length of RM? 7x M R 2x + 25 N P

  14. Your Turn What is the length of FD? 6x + 3 B F 4x + 9 A D

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