Math Section 4.4

Equidistance Theorems

By: Dan Schlosser and Brad Wagner

If 2 points are the same distance from a 3rd point they are said to be equidistant from that point.

AC

B

AB is equidistant to CB.

Look at these 2 figures.

AND

A

D

NS

B

R

A

DWW

In both cases the white line is the perpendicular bisector of the black

line.

The definition of a perpendicular bisector is a line that bisects and is perpendicular to another line.

Theorems that deal with perpendicular bisectors are:

Theorem 24: if 2 points are each equidistant from the end-points of a segment, then the 2 points determine the perpendicular bisector of that segment.

Theorem 25: if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoint of that segment.

Sample problem using theorem 24A

B D

E

Given: AB AD=~

AE BD

Prove: AE is the perpendicular bisector of BD

EAD EAB=~

Answer:

Statement Reason

1. AB AD=~ 1. Given

2. AE BD 2. Given

3. AED and AEB are right s

=~4. AED AEB

5. AE AE=~

6. EAD EAB~=

7. EAD EAB=~

8. BE DE=~

9. AE bisector of BD

3. lines form right s

4. right are=~

5. reflexive

6. Given

7. ASA (4, 5, 6)8. CPCTC

9.if 2 points are each equidistant from the end-points of a segment, then the 2 points determine the perpendicular bisector of that segment.

Sample problem using theorem 25

W

X Y

Z

Given: WZ is the bisector of XY.

Prove: WXZ WYZ=~

Answer:

Statement Reason

1. WZ bisector of XY 1. Given

2. WX WY=~ 2. if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoint of that segment.3. XZ YZ=~

3. same as 24. WZ WZ=~

4. reflexive5. WXZ WYZ=~

5. SSS (2, 3, 4)6. WXZ WYZ=~

6. CPCTC

Practice problem

A

B C

D

Given: AD is the bisector of XY and

Find: the perimeter of ABC

AC = 12, BD = 5

Answer:

If BD=5 then to get the length of BC you must do 5x2 to get BC=10. Then since AC=12, AB=12 because AD is the bisector of BC. Therefore the perimeter of ABC is 34 because 10+12+12=24.

Practice problem

M N

O

P

Given: PO is the bisector of MN and

PNO = 70

Find: m MPN

Answer:

If PNO=70 then PMO=70 because PO is perpendicular bisector of MN. 70 + 70 = 140 and since we know a triangles angles must equal 180, 180 – 140 = 40. Which means MPN = 40.

Works Cited

Milauskas, George, and Robert Whipple. Geometry forEnjoyment and Challenge. Boston: Houghton Mifflin Company1991.Print.