Table of Contents Date: Topic: Description: Page:

Section 5.1 : Bisectors of Triangles

Perpendicular Bisector: Vocabulary:   Segment Bisector:  A segment that cuts another segment into two congruent halves Perpendicular Bisector: A segment bisector that ALSO cuts the segment at a 90 angle.

Vocabulary: perpendicular bisector of a segment, then that point is    Perpendicular Bisector Theorem:  If a point is on the perpendicular bisector of a segment, then that point is equidistant from the endpoints of the bisected segment. Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of a segment.

Example 1: a) Find the length of BC. What type of relationship should I use in order to find the length of BC?

Example 1: b) Find the length of XY. How is this diagram different to the previous diagram?

Example 1: c) Find the length of PQ.

Vocabulary: intersect at a common point. concurrent lines intersect.    Concurrent Lines:  When 3 or more lines intersect at a common point. Point of Concurrency:  The point where the concurrent lines intersect.

Vocabulary: of a triangle intersect at a point called the circumcenter    Circumcenter Theorem  The perpendicular bisectors of a triangle intersect at a point called the circumcenter that is equidistant from the vertices.

Example 2: A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden?

Angle Bisector Theorem: Vocabulary:    Angle Bisector  A segment that cuts an angle into two halves. Angle Bisector Theorem:  If a point on the bisector of an angle, then it is equidistant from the sides of the angle.   

Converse of the Angle Bisector Theorem: Vocabulary:    Converse of the Angle Bisector Theorem:  If a point in the interior of an angle is equidistant from the sides of an angle, then it is an angle bisector.

Example 3: a) Find the length of DB.   What conclusion can be made if given a segment is an angle bisector?

Example 3: b) Find measure of angle WYZ. What conclusions can be made if given the two segments from the angle bisector are equal and intersect at a right angle?

Example 3: c) Find the length of QS. Given the set up to the right, what can I do to my two expressions? And why?

Vocabulary: The point of concurrency Incenter:   Incenter:  The point of concurrency of the Angle bisectors of a triangle. Incenter Theorem:  The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the triangle.

Example 4: Find ST if S is the incenter of ΔMNP. Find mSPU if S is the incenter of ΔMNP. If S is the incenter, what do I know about my triangle?

Summary! 1) Find the value of x. 2) Find the length of KL.

Summary! 3) Find 𝑚∡𝐿𝑌𝐹 4) Point A is the incenter of ∆𝑃𝑄𝑅. Find the measure of ∡𝑄𝑃𝐾.