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Understanding Equidistance and the Perpendicular Bisector in Geometry

This chapter delves into the concepts of distance, equidistance, and perpendicular bisectors in geometry. It defines distance as the shortest path between two objects and presents the postulate that a line segment is the shortest path between two points. The definition of equidistance is explored, stating that if two points are equal distance from a third point, they are equidistant. The chapter also introduces the Perpendicular Bisector Theorem, outlining how to determine a perpendicular bisector of a segment, and discusses properties and theorems related to equidistant points.

neil-stephenson
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Understanding Equidistance and the Perpendicular Bisector in Geometry

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  1. EQUIDISTANCE Geometry Chapter 4.4

  2. Some basics: A.) (def) Distance (between two objects) is the length of the shortest path joining them. B.) (postulate) A line segment is the shortest path between two points.

  3. Definition of Equidistance: • If 2 points P and Q are the same distance from a third point X, then X is said to be _____________from P and Q. EQUIDISTANT

  4. Definition of a Perpendicular Bisector: • A perpendicular bisector of a segment is the line that both _________ and is _____________to the segment. BISECTS PERPENDICULAR

  5. Theorem 24: If 2 points are equidistant from the endpoints of a segment, then they determine the perpendicular bisector of the segment.Abbrev: (PBT)(Perpendicular Bisector Theorem) NEEDED: 2 points equidistant or 2 pairs congruent segments

  6. 1. 2. 3. 4. 5. 6. Given CPCTC (1) Def of =Dist (2) CPCTC (1) Def of =Dist (4) If 2 points are =dist from the endpoints of a segment, then they determine the perpendicular bisector. (3,5) Given: Prove: 6. PBT

  7. Theorem 25: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Abbrev: (PBP) (Perpendicular Bisector Property) NEEDED: Perpendicular Bisector of the Segment.

  8. 1. 2. 3. X is =dist from A & B 4. 5. F is =dist from A & B 6. 7. 1. Given 2. Reflexive 3.If a point is on the perp bisector, then it is =dist. (1) 4. Def of =Dist (3) 5. Same as #3 (1) 6. Def of =Dist (5) 7. SSS (2,4,6) Given:Prove: S 3. PBP S S

  9. TRUE/FALSEPRACTICEReady??

  10. 1. E is the midpoint of BC. TRUE

  11. 2. <AEC is a right angle TRUE

  12. 3. E is the midpoint of AD FALSE

  13. 4. TRUE

  14. 5. TRUE

  15. 6. FALSE

  16. 7. FALSE

  17. 8. FALSE

  18. Prove the following statement: The line drawn from the vertex angle of an isosceles triangle through the point of intersection of the medians to the legs is perpendicular to the base.

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