5.1 & 5.2 Perpendiculars & Bisectors

Objectives: - Use properties of perpendicular bisectors - Use properties of angle bisectors to find distances - Use bisectors in a triangle

Perpendicular Bisectors

A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector.

Line CP is a bisector of segment AB

P

A

B

C

Perpendicular Bisectors

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

In this diagram, if P is the midpoint of AB, then P is equidistant from A and B

P

A

B

C

Perpendicular Bisector Theorem

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

If CP is the perpendicular bisector of AB, then CA = CB

P

A

B

C

Converse of the Perpendicular Bisector Theorem

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

If CA = CB then C lies on the perpendicular bisector of AB.

P

A

B

C

Show MN is the perpendicular bisector of ST

What segment lengths in the diagram are equal?

NS and NT, because MN bisects STMS and MT, by the theorem we just learnedQS and QT, because they are both 12

N

T

S

M Q

12

12

Show MN is the perpendicular bisector of ST

Explain why Q is on MNQS = QT, so Q is equidistant from S and T. By

the converse theorem we just learned, Q is on the perpendicular bisector of ST, which is MN.

N

T

S

M Q

12

12

Properties of Angle Bisectors

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

For example, the distance between Q and line m is QP

P m

Q

Properties of Angle Bisectors

When a point is the same distance from one line as it is from another line, then the point is equidistant from the two lines (or rays or segments).

Angle Bisector Theorem

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

If mBAD = mCAD, then DB = DC

A

B

D

C

Converse of 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.

If DB = DC, then mBAD = mCAD

A

B

D

C

Look at p. 267 Example 3

You are given that B bisects CAD and that ACB and ADB are right angles. What can you say about BC and BD?

Because BC and BD meet AC and AD at right angles, they are perpendicular segments to the sides of CAD.

This implies that their lengths represent the distances from the point B to AC and AD.

Because point B is on the bisector of CAD, it is equidistant from the sides of the angle.

So, BC = BD and you can conclude that segment BC ~= segment BD.

Do p. 267 1-7

Using Perpendicular Bisectors of a Triangle

A perpendicular bisector of a triangle is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.

Concurrent lines

When 3 or more lines intersect in the same point, they are called concurrent lines. The point of intersection of the lines is called the point of concurrency.

Concurrent lines

The 3 perpendicular bisectors of a triangle are concurrent. The point of concurrency can be inside, on, or outside the triangle.

The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle.

Concurrency of Perpendicular Bisectors of a Triangle Theorem

The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.

See picture on p. 273

Useful Stuff!

See page 273, Example 1

Using Angle Bisectors of a Triangle

An angle bisector of a triangle is a bisector of an angle of the triangle.

The 3 angle bisectors are concurrent. The point of concurrency is called the

incenter of the triangle and always lies inside the triangle.

Concurrency of Angle Bisectors of a Triangle Theorem

The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

See picture on page 274.

Look at Example 2, p. 274

The angle bisectors of ∆ MNP meet at point L.

What segments are congruent?By the theorem we just learned, the 3

angle bisectors of a triangle intersect at a point that is equidistant from the sides of a triangle.

So, LR ~= LQ ~= LS

Look at Example 2, p. 274

Find LQ and LRUse the Pythagorean Theorem to find LQ in

∆LQM (LQ)2 + (MQ)2 = (LM)2

(LQ)2 + 152 = 172

(LQ)2 + 225 = 289 (LQ)2 = 64LQ = 8Because LQ ~= LR, LR also = 8

Do p. 275 1-4

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