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:

1. To use and define perpendicular bisectors, angle bisectors, medians, and altitudes

2. To discover, use, and prove various theorems about perpendicular bisectors and angle bisectors

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.

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.

Example 1

Plan a proof for the Perpendicular Bisector Theorem.

1. AB CP Given2. AP = PB Def. of Bisector 3. <CPB & Def. of Perp. <CPA R rt. <s 4. <CPB=<CPA All rt. <s R =5. CP = CP Reflexive6. CAP = CBP SAS7. CA = CB CPCTC

~

~ ~~

~~

Example 2

BD is the perpendicular bisector of AC. Find AD.

X=7AD=35

Example 3

Find the values of x and y.

x=12y=8

Angle Bisector

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

Angle Bisector Theorem

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

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

Example 5

Find the value of x.

x=5

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.

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.

Example 7

For what value of x does P lie on the bisector of <A?

x=4

Special Triangle Segments

Both perpendicular bisectors and angle bisectors are often associated with triangles, as shown below. Triangles have two other special segments.

A

B

C

B

A C

Angle Bisector

Pe

rpe

nd

icu

lar

Bis

ec

tor

Median

Median

A median of a triangle is a segment from a vertex to the midpoint of the opposite side of the triangle.

Altitude

Altitude

The length of the altitude is the height of the triangle.

An altitude of a triangle is a perpendicular segment from a vertex to the opposite side or to the line that contains that side.

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?