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4.4 The Equidistance Theorems

4.4 The Equidistance Theorems. Objective: After studying this lesson you will be able to recognize the relationship between equidistance and perpendicular bisection. Definition The distance between two points is the length of the shortest path joining them.

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4.4 The Equidistance Theorems

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  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. 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. A A A C D 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 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. 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. Given Given Reflexive Property SSS (1,2,3) CPCTC Reflexive Property SAS (1,5,6) CPCTC If a line divides a segment into 2 congruent segments, it bisects it. CPCTC 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

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

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

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

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

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