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5.2: Bisectors in Triangles

This lesson focuses on understanding the properties of perpendicular and angle bisectors in triangles. Students will learn about the definitions of perpendicular bisectors and angle bisectors, and the concept of equidistance. Key theorems, including the Perpendicular Bisector Theorem and Angle Bisector Theorem, will be explored through practical examples and problem-solving exercises. Students will practice finding points and segment lengths, reinforcing their understanding of these essential geometric concepts.

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5.2: Bisectors in Triangles

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  1. 5.2: Bisectors in Triangles Objectives: To use properties of perpendicular and angle bisectors

  2. Warm Up • What is a perpendicular bisector of a segment? • What is an angle bisector? • What does equidistant mean?

  3. is the perpendicular bisector of What do we know?

  4. Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

  5. Converse of Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. IS THE PERPENDICULAR BISECTOR OF SEGMENT AB 6 6

  6. EXAMPLES • Find PB and AQ. 14 7

  7. Find AD, x, and BC. 12 C D A 2x+6 3x+1 B

  8. What do we know about P? 10 10

  9. Definition The distance from a point to a line is defined as the length of the perpendicular segment from the point to the line.

  10. Angle Bisector Theorem • If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. 4 4

  11. Converse of Angle Bisector Theorem • If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

  12. Let’s try an example… • Solve for x. Then find AD & CD. 5x = 2x + 24 3x = 24 X = 8 AD=5*8=40 CD=2*8+24=40 5x 2x + 24

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