Perpendiculars and Bisectors I

Perpendicular Bisector

• A line, ray, segment or plane perpendicular to a segment at its midpoint.

is a ⊥ bisector of

Constructions• Construct a perpendicular bisector to a line

segment .

• Construct a perpendicular to a line l, through a point P on l.

Equidistant

• A point is equidistant from two points if its distance from each point is the same.

• C is equidistant from A and B, since C was drawn so that CA = CB.

Theorem: Perpendicular Bisector

• If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Proof of the Perpendicular Bisector Theorem

Given:

bisects Prove:Show ∆ACP ≅

∆BCP

CA = CB

bisects ∠APC ≅∠CPB

Reflexive property of congruence∆ACP ∆≅ BCP SAS Postulate

CPCTCCA = CB Definition of Congruence

Plan:

Theorem: Perpendicular Bisector Converse

• If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

If DA = DB, then D lies on the perpendicular bisector of

Prove the Perpendicular Bisector Theorem Converse

Given: C is equidistant from A and B

Prove: C is on the perpendicular bisector of

Plan

DrawShow ∆APC ∆≅ BPC

Example

Homework

• Exercise 5.1 page 267: 1-10, 14, 16-18, 29.