1. BY: Kyle Bormann, Matt Heckman, and Ryan Gilbert Equidistance Theorems Section 4.4

2.  Definition: The distance between two objects 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 C are the same distance from a third point B, then B is said to be equidistant from A and C. A B C

3. What is a Perpendicular Bisector?  The perpendicular bisector of a segment is the line that bisects and is perpendicular to the segment.  Example: M A T H

4.  Theorem 24: If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.  Example: R YA N D R YA N D

5. Theorem 25  Theorem 25: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints. K Y LE R K Y L E R

6. If AB ≅ CB and AD ≅ DC, then DB is the perpendicular bisector of AC (If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment. A B C D E

7. Given: AM ≅ MH AP ≅ PH Example Problem 2 M A H T P Prove: ΔAPT ≅ ΔHPT

8. Statements Reasons 1. AM ≅ MH 1. Given 2. AP ≅ PH 2. Given 3. MT is the perpendicular bisector 3.If two points are each equidistant from the endpoints of a of AH segment, then the two points determine the perpendicular bisector 4. < PTH, <PTA are right angles 4. Perpendicular lines form right angles 5. ΔPTH, ΔPTA are right triangles 5. If a triangle contains a right angle, then it is a right triangle 6. PT ≅ PT 6. Reflexive 7. ΔPTH ≅ ΔPTA 7. HL (2,5,6) of that segment

9. Example Problem 3 K Y L E B Given: KL is the perpendicular bisector of YE Prove: ΔKLE ≅ ΔKLY

10. Statements Reasons 1. KL is the perpendicular 1. Given bisector of YE 2. YB ≅ BE 2. If a point is on the perpendicular bisector of a 3. KY ≅ KE 3. Same as 2 4. KB ≅ KB 4. Reflexive 5. ΔKBY ≅ ΔKBE 5. SSS (2,3,4) 6. <LKE ≅ <LKY 6. CPCTC 7. KL ≅ KL 7. Reflexive 8. ΔKLE ≅ ΔKLY 8. SAS (3,6,7) segment, then it is equidistant from the endpoints of that segment

11. Works Cited  Milauskas, George; Rhoad, Richard; Whipple, Robert. Geometry for Enjoyment and Challenge. Illinois: McDougal Littell,1991. Print.