5-2 Bisectors of a Triangle Rigor: apply the perpendicular bisector theorem, the angle bisector theorem, and their converses Relevance: City planning and interior design

Exploring Perpendicular Bisectors in a triangle Draw a triangle (any triangle) on a piece of tracing paper Fold each side in half. The creases are the perpendicular bisectors of each side. What do you notice? Concurrent – 3 or more lines intersect at one point: the point of concurrency.

Circumcenter – the point of concurrency of the ┴ bisectors Concurrency of Perpendicular Bisectors Theorem – the circumcenter of a ∆ is equidistant from the vertices AP = BP = CP Circumcenter can be inside, outside, or on ∆

Circumscribed Circles Circumscribed circle – a circle that has all 3 vertices of a triangle on the circle with the center of circle as the circumcenter of the triangle The prefix circum- means “around”, so a circumscribed circle goes around the triangle Turn in your core book to page 199 EX 1 and construct a circumscribed circle

EX 1: What are the coordinates of the circumcenter of ∆ with vertices A(2,7), B(10,7) & C(10,3) Step 1: Graph ∆ Step 2: Calculate midpoints (count if vertical or horizontal sides) Step 3: Use right angle to draw ┴ bisector

EX 2: City Planning

Angle Bisectors and Incenters Inscribed circle – a circle that touches every side of the triangle with the incenter as its center Turn to pg 200 in the core book and construct an inscribed circle

Concurrency of Angle Bisectors Theorem Incenter – the point of concurrency of the angle bisectors; always inside the triangle Theorem: The incenter of a triangle is equidistant from the sides to the triangle.

EX 3: Calculate the length of the segment. A) GE = 2x – 7; GF = x + 4 What is GD? B) QN = 5x + 36; QM = 2x + 51 What is QO?

EX 4: Camping Logan plans to go camping in a state park. The park is bordered by 3 highways, and Logan wants to pitch his tent as far away from the highways as possible. Should he set up camp at the circumcenter or the incenter of the park? Why?

5-2 Classwork Core book pgs 201 – 202 #1 – 3, 7 – 9 Textbook pg 323 – 324 #9, 10, 14 – 16, 20, 22, 23, 26 – 32 5-2 Homework Core book pg 203 – 204 ALL Due Thursday for periods 1, 3, 5 Due Friday for periods 2, 4, 7