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

2. DefinitionThe distance between two points is the length of the shortest path joining them. PostulateA 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. B A Z

3. B C A D B C A D B C A D 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. DefinitionThe perpendicular bisector of a segment is the line that bisects and is perpendicular to the segment. TheoremIf 2 points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.

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

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

7. 1. 2. 3. 4. 5. Given: Prove: 1. 2. 3. 4. 5. B C E A D 1 3 2 4 If 2 points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.

8. 1. 2. 3. 4. Given: Prove: 1. 2. 3. 4. BC E A If 2 points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.

9. 1. 2. 3. 4. Given: Prove: 1. 2. 3. 4. B C E A D A If 2 points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment. A point of the perpendicular bisector of a segment is equidistant from the endpoints of the segment

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