Perpendiculars and Angle Bisectors Skill 27 Objective HSGCO

Perpendiculars and Angle Bisectors Skill 27

Objective HSG-CO. 9: Students are responsible for using properties of perpendicular bisectors, and using these properties to prove theorems.

Definitions The distance from a point on a line is the length of the perpendicular segment from the point to the line. A point is equidistant from two objects if it is the same distance from the two objects.

Theorem 24: 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 A M B

Theorem 25: Converse Perpendicular Bisector Theorem If a point is equidistant from the end points of a segment, then it is the on the perpendicular bisector of the segment. P A If ���� =���� M B

Theorem 26: Angle Bisector Theorem If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. P Q S M

Theorem 27: Converse Angle Bisector Theorem If a point is equidistant from the sides of the angle, then the point is on the angle bisector. P Q S M

Example 1; Using the Perp. Bisector Theorem A D B B is equidistant from A and C C

Example 2; Using the Perp. Bisector Theorem A park director wants to build a T-shirt stand equidistant from the Roller Coaster, R, and the Space Shot, S. What are the possible locations of the stand? Explain. P R Find the distance from R to S. S Draw a line with a 90ᵒ angle at that point. The t-shirt stand can be placed anywhere on that line. M

Example 3; Using the Angle Bisector Theorem R is equidistant from P and M M R N P

Example 4; Using the Angle Bisector Theorem F is equidistant from B and D B C F D

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