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