2.5.6 Perpendicular and Angle Bisectors
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The document outlines concepts related to perpendicular and angle bisectors, including their construction and applications in geometry. It explains the theorems related to these bisectors, such as the perpendicular bisector theorem and angle bisector theorem, along with their converses. Additionally, it defines the circumcenter and incenter of a triangle as key points derived from these bisectors.
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2.5.6 Perpendicular and Angle Bisectors
- 1. Perpendicular and Angle Bisectors The student is able to (I can): • Construct perpendicular and angle bisectors • Use bisectors to solve problems • Identify the circumcenter and incenter of a triangle
- 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. P D A E PD = AD PE = AE
- 3. Converse of Perp. Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. S K YT ST = YT
- 4. Examples Find each measure: 1. YO YO = BO = 15 2. GR B O Y 15 G I R L 20 20 2x-1 x+8 2x — 1 = x + 8 x = 9 GR = 2x — 1 + x + 8 = 34
- 5. Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Converse of the Angle Bisector Theorem If a point is equidistant from the sides of an angle, then it is on the angle bisector. A L G N AN = GN ∠ALN ≅ ∠GLN
- 6. circumcenter The intersection of the perpendicular bisectors of a triangle.
- 7. circumcenter The intersection of the perpendicular bisectors of a triangle. It is called the circumcenter, because it is the center of a circle that circumscribescircumscribescircumscribescircumscribes the triangle (all three vertices are on the circle).
- 8. incenter The intersection of the angle bisectors of a triangle.
- 9. incenter The intersection of the angle bisectors of a triangle. It is called the incenter because it is the center of the circle that is inscribedinscribedinscribedinscribed in the circle (the circle just touches all three sides).