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5.2 bisectors of a triangle

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5.2 bisectors of a triangle

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

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