5.1 perpendiculars and bisectors i
TRANSCRIPT
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Perpendiculars and Bisectors I
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Perpendicular Bisector
• A line, ray, segment or plane perpendicular to a segment at its midpoint.
is a ⊥ bisector of
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Constructions• Construct a perpendicular bisector to a line
segment .
• Construct a perpendicular to a line l, through a point P on l.
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Equidistant
• A point is equidistant from two points if its distance from each point is the same.
• C is equidistant from A and B, since C was drawn so that CA = CB.
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Theorem: Perpendicular Bisector
• If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
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Proof of the Perpendicular Bisector Theorem
Given:
bisects Prove:Show ∆ACP ≅
∆BCP
CA = CB
bisects ∠APC ≅∠CPB
Reflexive property of congruence∆ACP ∆≅ BCP SAS Postulate
CPCTCCA = CB Definition of Congruence
Plan:
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Theorem: Perpendicular Bisector Converse
• If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
If DA = DB, then D lies on the perpendicular bisector of
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Prove the Perpendicular Bisector Theorem Converse
Given: C is equidistant from A and B
Prove: C is on the perpendicular bisector of
Plan
DrawShow ∆APC ∆≅ BPC
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Example
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Homework
• Exercise 5.1 page 267: 1-10, 14, 16-18, 29.