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5.2 bisectors of a triangle

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Robert Hammarstrand

The perpendicular bisectors and angle bisectors of a triangle intersect at points that are equidistant from the triangle's vertices and sides, respectively. The perpendicular bisectors intersect at the triangle's circumcenter, which is equidistant from the vertices. The angle bisectors intersect at the triangle's incentre, which is equidistant from the sides. These properties are proven using theorems about congruent triangles and corresponding parts of congruent triangles.

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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
油
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 = CP Plan: 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 = PF Plan: 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

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5.2 bisectors of a triangle

  • 1. Bisectors of a Triangle
  • 2. Perpendicular Bisector A line, ray or segment that is perpendicular to a side of a triangle at the midpoint of the side.
  • 3. 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.
  • 5. 油
  • 6. 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
  • 7. 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
  • 9. 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
  • 10. 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
  • 12. Summary of Vocabulary Perpendicular Bisector Angle Bisector Concurrent Lines Circumscribe Circumcentre Incentre
  • 13. Proof of Concurrency of 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 Concurrency of 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 5.2 page 275: 1-39, odd. Workbook 5.1, 5.2 Collect workbooks Monday