5.2 bisectors of a triangle
TRANSCRIPT
![Page 1: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/1.jpg)
Bisectors of a Triangle
![Page 2: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/2.jpg)
Perpendicular Bisector
• A line, ray or segment that is perpendicular to a side of a triangle at the midpoint of the side.
![Page 3: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/3.jpg)
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.
![Page 4: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/4.jpg)
Perpendicular Bisectors of a Triangle
![Page 5: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/5.jpg)
![Page 6: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/6.jpg)
• 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
![Page 7: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/7.jpg)
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
![Page 8: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/8.jpg)
Example
![Page 9: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/9.jpg)
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
![Page 10: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/10.jpg)
• 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
![Page 11: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/11.jpg)
Example 2
![Page 12: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/12.jpg)
Summary of Vocabulary
• Perpendicular Bisector
• Angle Bisector
• Concurrent Lines
• Circumscribe
• Circumcentre
• Incentre
![Page 13: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/13.jpg)
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
![Page 14: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/14.jpg)
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
![Page 15: 5.2 bisectors of a triangle](https://reader034.vdocuments.mx/reader034/viewer/2022042518/5559caa3d8b42a93208b479c/html5/thumbnails/15.jpg)
Homework
• Exercise 5.2 page 275: 1-39, odd.• Workbook 5.1, 5.2• Collect workbooks Monday