5.2 bisectors of a triangle
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Transcript of 5.2 bisectors of a triangle
Bisectors of a Triangle
Perpendicular Bisector
• A line, ray or segment that is perpendicular to a side of a triangle at the midpoint of the side.
Concurrent Lines
• Concurrent lines (segments or rays) are lines which lie in the same plane and intersect in a single point. The point of intersection is the point of concurrency. For example, point A is the point of concurrency.
Perpendicular Bisectors of a Triangle
• Concurrent
• Point of concurrency may be inside or outside
• A circle may be circumscribed
• The point of concurrency is called the circumcentre
Perpendicular Bisectors of a Triangle
Theorem: Concurrency of Perpendicular Bisectors of a Triangle
• The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of a triangle.
• PA = PB = PC
Example
Angle Bisectors of a Triangle
• Bisects an angle of the triangle.
• Three angle bisectors
–concurrent
• The point of concurrency: incentre.
• The incentre is equidistant from the sides
• The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.
• PD = PE = PF
Theorem: Concurrency of Angle Bisectors of a Triangle
Example 2
Summary of Vocabulary
• Perpendicular Bisector
• Angle Bisector
• Concurrent Lines
• Circumscribe
• Circumcentre
• Incentre
Proof of Concurrency of Perpendicular Bisectors of a Triangle Theorem
Prove: AP = BP = CPPlan: • Show ∆ADP ∆≅ BDP• and ∆BPF ∆≅ CPF
Sketch: •∆ADP ∆≅ BDP (SAS)
• AP = BP (CPCTC)• ∆BPF ∆≅ CPF (SAS)
• BP = CP (CPCTC)• AP = BP = CP
Proof of Concurrency of Angle Bisectors of a Triangle Theorem
Prove: PD = PE = PFPlan: • Show ∆CDP ∆≅ CEP• and ∆AFP ∆≅ AEP
Sketch: •∆CDP ∆≅ CEP (AAS)
• PD = PE (CPCTC)• ∆AFP ∆≅ AEP (AAS)
• PE = PF (CPCTC)• PD = PE = PF
Homework
• Exercise 5.2 page 275: 1-39, odd.• Workbook 5.1, 5.2• Collect workbooks Monday