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Equidistant

Equidistant. A point is equidistant from two figures if the point is the same distance from each figure. Examples: midpoints and parallel lines. 5.2-5.4: Special Segments. Objectives: To use and define perpendicular bisectors, angle bisectors, medians, and altitudes

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Equidistant

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  1. Equidistant A point is equidistantfrom two figures if the point is the same distance from each figure. Examples: midpoints and parallel lines

  2. 5.2-5.4: Special Segments Objectives: • To use and define perpendicular bisectors, angle bisectors, medians, and altitudes • To discover, use, and prove various theorems about perpendicular bisectors and angle bisectors

  3. 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.

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

  5. Example 1 Plan a proof for the Perpendicular Bisector Theorem. 1. AB CP Given 2. AP = PB Def. of Bisector 3. <CPB & Def. of Perp. <CPA R rt. <s 4. <CPB=<CPA All rt. <s R = CP = CP Reflexive CAP = CBP SAS CA = CB CPCTC ~ ~ ~ ~ ~ ~

  6. Example 2 BD is the perpendicular bisector of AC. Find AD. X=7 AD=35

  7. Example 3 Find the values of x and y. x=12 y=8

  8. Angle Bisector An angle bisector is a ray that divides an angle into two congruent angles.

  9. Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

  10. Example 4 A soccer goalie’s position relative to the ball and goalposts forms congruent angles, as shown. Will the goalie have to move farther to block a shot toward the right goal post or the left one? Answer in your notebook

  11. Example 5 Find the value of x. x=5

  12. Example 6 Find the measure of <GFJ. It’s not the Angle Bisector Theorem that could help us answer this question. It’s the converse. If it’s true.

  13. Converse of the Angle Bisector Theorem If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.

  14. Example 7 For what value of x does P lie on the bisector of <A? x=4

  15. Special Triangle Segments Both perpendicular bisectors and angle bisectors are often associated with triangles, as shown below. Triangles have two other special segments. Perpendicular Bisector Angle Bisector

  16. Median

  17. Median A medianof a triangle is a segment from a vertex to the midpoint of the opposite side of the triangle.

  18. Altitude

  19. Altitude The length of the altitude is the height of the triangle. An altitudeof a triangle is a perpendicular segment from a vertex to the opposite side or to the line that contains that side.

  20. Example 8 Is it possible for any of the aforementioned special segments to be identical? In other words, is there a triangle for which a median, an angle bisector, and an altitude are all the same?

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