1. Lesson 5-1: Perpendicular & Angle Bisectors
Rigor: prove the perpendicular bisector theorem, the angle bisector theorem, and their converses
Relevance: getting ready for the chapter 4 test!

2. A second look at a few theorems. It’s time to prove them!
SAS
CPCTC
SSS
CPCTC + definition of ┴

3. The distance from a point to a line is the length of the perpendicular segment from the point to the line.

4. AAS
CPCTC
HL
CPCTC + Definition of a bisector

5. Note the difference…
An angle bisector contains all the points equidistant from 2 SIDES.
A perpendicular bisector contains all of the points equidistant from 2 POINTS.

6. Lesson 5-2: Bisectors or Triangles
Rigor: Apply the perpendicular bisector theorem, the angle bisector theorem, and their converses to triangles
Relevance: City planning and interior design

7. Exploring Perpendicular Bisectors in a triangle
Trace PQR from workbook pg 199 to the middle of 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.

8. 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 ∆

9. Exploring Perpendicular Bisectors in a triangle
Transfer the circumcenter from your tracing paper to the triangle in your workbook on pg 199
Set your compass from the circumcenter to any vertex of PQR.
Construct a circle around the triangle.
What do you notice?

10. 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 is drawn around the triangle!

11. Drawing triangles circumscribed by circles.

12. EX 1: City Planning

13. EX 2: What are the coordinates of the circumcenter of ∆ with vertices A(2,7), B(10,7) & C(10,3)
Step 1: Graph ∆ on graph paper
Step 2: Trace the triangle with tracing paper.
Step 3: fold the sides in half (┴ bisectors) to locate circumcenter
Step 4: Overlay tracing paper on graph
Constructing the ┴ bisectors is okay too!

14. Exploring Angle Bisectors in a triangle
Trace PQR from workbook pg 200 to the middle of a piece of tracing paper
Fold each angle in half. The creases are the angle bisectors of each angle.
What do you notice?

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

16. Angle Bisectors and Incenters
Inscribed circle – a circle that touches every side of the triangle once with the incenter as its center
“inscribe” means to be drawn inside!
Turn to pg 200 in the workbook & construct an inscribed circle

17. EX 3: Camping
Brandon plans to go camping in a state park. The park is bordered by 3 highways, and Brandon 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?

18. Recap… ANGLE bisectors of a triangle are equidistant from the SIDES.
Intersect at the incenter that is the center of an inscribed circle inside the triangle
Perpendicular bisectors of a triangle cut the SIDES in half and are equidistant from the ANGLES.
Intersect at the circumcenter that is the center of a circumscribed circle going around the triangle

19. 5-2 Assignment Workbook pg 203 – 204 ALL Sudo-constructions are fine
Use tracing paper to locate the circumcenter or incenter (unless you want the challenge of using a compass!)
Use a compass to construct the circle