5.2 bisectors of a triangle

5.2 bisectors of a triangle

• 1.
  Bisectors of aTriangle
• 2.
  Perpendicular Bisector Aline, ray or segment that is perpendicular to a side of a triangle at the midpoint of the side.
• 3.
  Concurrent Lines Concurrentlines (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.
• 4.
  Perpendicular Bisectors ofa Triangle
• 5.
   
• 6.
  Concurrent Point ofconcurrency may be inside or outside A circle may be circumscribed The point of concurrency is called the circumcentre Perpendicular Bisectors of a Triangle
• 7.
  Theorem: Concurrency ofPerpendicular 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
• 8.
  Example
• 9.
  Angle Bisectors ofa Triangle Bisects an angle of the triangle. Three angle bisectors concurrent The point of concurrency: incentre . The incentre is equidistant from the sides
• 10.
  The angle bisectorsof 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
• 11.
  Example 2
• 12.
  Summary of VocabularyPerpendicular Bisector Angle Bisector Concurrent Lines Circumscribe Circumcentre Incentre
• 13.
  Proof of Concurrencyof Perpendicular Bisectors of a Triangle Theorem Prove: AP = BP = CP Plan: Show ∆ ADP ≅ ∆ BDP and ∆ BPF ≅ ∆ CPF Sketch : ∆ ADP ≅ ∆ BDP (SAS) AP = BP (CPCTC) ∆ BPF ≅ ∆ CPF (SAS) BP = CP (CPCTC) AP = BP = CP
• 14.
  Proof of Concurrencyof Angle Bisectors of a Triangle Theorem Prove: PD = PE = PF Plan: Show ∆ CDP ≅ ∆ CEP and ∆ AFP ≅ ∆ AEP Sketch : ∆ CDP ≅ ∆ CEP (AAS) PD = PE (CPCTC) ∆ AFP ≅ ∆ AEP (AAS) PE = PF (CPCTC) PD = PE = PF
• 15.
  Homework Exercise 5.2page 275: 1-39, odd. Workbook 5.1, 5.2 Collect workbooks Monday