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

( , )2 2

x x y yM

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: CampingLogan 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