DWG NO. 05.2 — Lesson 2 of 5

Perpendicular and Angle Bisectors

Unit 5: Relationships Within Triangles · ~15–30 min

Objective

Apply the Perpendicular Bisector Theorem and the Angle Bisector Theorem, and their converses, to find missing distances.

Perpendicular bisectors

A perpendicular bisector of a segment passes through its midpoint at a 90° angle. Any point sitting on that line has a special guarantee about its distance to the segment's two endpoints.

Perpendicular Bisector Theorem: If a point lies on the perpendicular bisector of a segment, then it is equidistant from the segment's two endpoints.
Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the two endpoints of a segment, then it lies on the perpendicular bisector of that segment.

P lies on the perpendicular bisector of AB, so PA = PB.

Angle bisectors

An angle bisector splits an angle into two congruent halves. A point on that ray has a matching guarantee about its distance to the angle's two sides — measured as the shortest, perpendicular distance to each side.

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

P lies on the bisector of angle V, so its perpendicular distance to each side is equal.

Worked Example 1 · Perpendicular bisector

ProblemPoint P lies on the perpendicular bisector of segment AB. If PA = 4x + 3 and PB = 27, find x.

1.By the Perpendicular Bisector Theorem, PA = PB.

2.4x + 3 = 27, so 4x = 24, giving x = 6.

x = 6

Worked Example 2 · Angle bisector

ProblemRay VP bisects angle UVW. Point P is 9 units from side VU. How far is P from side VW?

1.By the Angle Bisector Theorem, a point on the bisector is equidistant from both sides of the angle.

2.So the distance from P to VW must equal the distance from P to VU.

9 units

Guided practice

1.Point Q is on the perpendicular bisector of segment CD. If QC = 15, find QD.
Show answer
QD = QC = 15, by the Perpendicular Bisector Theorem.
2.Point M is equidistant from points R and S. What can you conclude about M?
Show answer
By the Converse of the Perpendicular Bisector Theorem, M lies on the perpendicular bisector of segment RS.
3.Ray BD bisects angle ABC. Point D is 6.5 units from ray BA. How far is D from ray BC?
Show answer
6.5 units, by the Angle Bisector Theorem — distances to both sides are equal.
4.PA = 2x + 5 and PB = 3x − 1, where P lies on the perpendicular bisector of AB. Find PA.
Show answer
Set 2x + 5 = 3x − 1, so x = 6. Then PA = 2(6) + 5 = 17.
5.A point is the same perpendicular distance from both sides of angle T. What can you conclude, and by which theorem?
Show answer
The point lies on the bisector of angle T, by the Converse of the Angle Bisector Theorem.

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