Lesson 5 1 Perpendicular Angle Bisectors Rigor prove

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!

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

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

AAS CPCTC HL CPCTC + Definition of a bisector

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

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

Exploring Perpendicular Bisectors in a triangle Trace PQR from workbook pg 199 to the middle of a piece of tracing paper n Fold each side in half. The creases are the perpendicular bisectors of each side. n What do you notice? n q Concurrent – 3 or more lines intersect at one point: the point of concurrency.

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

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

Circumscribed Circles n 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 n The prefix circum- means “around”, so a circumscribed circle is drawn around the triangle!

Drawing triangles circumscribed by circles.

EX 1: City Planning

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 n Step 2: Trace the triangle with tracing paper. n Step 3: fold the sides in half (┴ bisectors) to locate circumcenter n Step 4: Overlay tracing paper on graph n Constructing the ┴ bisectors is okay too! n

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

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

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

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?

Recap… n 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 n 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

5 -2 Assignment n Workbook n 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