4.4 The Equidistance Theorems

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Presentation transcript:

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

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

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.

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.

B Given: E C A Prove: D 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. If a line divides a segment into 2 congruent segments, it bisects it. If 2 angles are both supplementary and congruent , then they are right angles. If 2 lines intersect to form right angles they are perpendicular. Combination of steps 9 and 12

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.

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

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

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

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