1. 4.4 The Equidistance Theorems
Objective:
After studying this lesson you will be able to recognize the relationship between equidistance and perpendicular bisection.

2. Postulate A line segment is the shortest path between two points
Definition The distance between two points is the length of the shortest path joining them.
Postulate A line segment is the shortest path between two points
If two points A and B are the same distance from a third point Z, then Z is said to be equidistant to A and B. Z B A

3. What do these drawings have in common?B C D C D B B
What do these drawings have in common?
A and B are equidistant from points C and D. We could prove that line AB is the perpendicular bisector of segment CD with the following theorems.

4. Definition. The perpendicular bisector of
Definition The perpendicular bisector of a segment is the line that bisects and is perpendicular to the segment.
Theorem If 2 points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.

5. Given:
Prove: 1. 2. 3. 4. 5. 6. 7. 8. 9.
10.
11.
12.
13. 1. 2. 3. 4. 5. 6. 7. 8. 9.
10.
11.
12.
13. B E C A D
If a line divides a segment into 2 congruent segments, it bisects it.
If 2 angles are both supp. and congruent , then they are rt. angles.
If 2 lines intersect to form right angles they are perpendicular.
Combination of steps 9 and 12

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

7. Given:
Prove: A 1. 2. 3. 4. 5. 1. 2. 3. 4. 5. 1 E 2 B D 3 4 C

8. Given: A
Prove: B C E 1. 2. 3. 4. 1. 2. 3. 4.

9. Given: A A C
Prove: B D 1. 2. 3. 4. 1. 2. 3. 4. E

10. Summary:
Define equidistant in your own words and summarize how we used it in proofs.
Homework: worksheet