**Homework: #1-6 from 5.1 worksheet** Agenda 11/8 Pass out chapter 5 workbooks Start Section 5.1 **Homework: #1-6 from 5.1 worksheet**

Section 5.1 Notes Part 1: Bisectors of Triangles EQ: What conjectures can you make about a point on the perpendicular bisector of a segment and a point on the bisector of an angle?

Perpendicular Bisector   Equidistant  Equal distance Segment Bisector A point, segment, line or plane that divides a line segment into two equal parts. Perpendicular Bisector If a bisector is also perpendicular to the segment, it is called a perpendicular bisector

If 𝐶𝐷 is a perpendicular bisector of 𝐴𝐵 , then AC = BC   Perpendicular Bisector Theorem  If 𝐶𝐷 is a perpendicular bisector of 𝐴𝐵 , then AC = BC Converse of the Perpendicular Bisector Theorem If AE = BE, then E lies on 𝐶𝐷 , the perpendicular bisector of 𝐴𝐵

Example 1: a) Find the length of BC. 𝐴𝐶 = 𝐵𝐶 𝐵𝐶 =8.5

Example 1: b) Find the length of XY. 𝑍𝑌 = 𝑋𝑌 𝑋𝑌 =6

Example 1: c) Find the length of PQ. 𝑃𝑄 = 𝑅𝑄 3𝑥+1=5𝑥−3 𝑥=2 𝑃𝑄 =7

Try the you try first before you look at the answer!

YOU TRY! 1. RS 2. EG 3. AD 𝑅𝑆 =6.8 𝐸𝐺 =19 3𝑥+14=5𝑥 𝑥=7 𝐴𝐷 =35

If 𝐹𝐷 ⊥ 𝐵𝐷 , 𝐹𝐸 ⊥ 𝐵𝐸 , and DF = FE, then 𝐵𝐹 bisects ∠𝐷𝐵𝐸   Angle Bisector Theorem  If 𝐵𝐹 bisects ∠𝐷𝐵𝐸, 𝐹𝐷 ⊥ 𝐵𝐹 , & 𝐹𝐸 ⊥ 𝐵𝐸 , then DF = FE Converse of the Angle Bisector Theorem  If 𝐹𝐷 ⊥ 𝐵𝐷 , 𝐹𝐸 ⊥ 𝐵𝐸 , and DF = FE, then 𝐵𝐹 bisects ∠𝐷𝐵𝐸

Example 2: a) Find the length of DB. 𝐷𝐵 =5

Example 2: b) Find m∠WYZ. ∠𝑊𝑌𝑍 = 68°

Example 2: c) Find the length of QS. 4𝑥−1=3𝑥+2 𝑥=3 𝑄𝑆 =11

YOU TRY! Find each measure. 𝑚∠𝐺𝐹𝐽   2. RS ∠𝐺𝐹𝐽 = 42° 5𝑥=6𝑥−5 𝑥=5 𝑅𝑆 =25

Section 5.1 Notes Part 2: Bisectors of Triangles EQ: What conjectures can you make about a point on the perpendicular bisector of a segment and a point on the bisector of an angle?

When three or more lines intersect at a common point   Concurrent Lines When three or more lines intersect at a common point Point of Concurrency    The point where concurrent lines intersect Circumcenter The point of concurrency of the perpendicular bisectors Circumcenter Theorem If P is the circumcenter of ∆ABC, then PB = PA = PC

Example 4 A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden? No, the circumcenter is outside of the garden.

If P is the incenter of ∆ABC, then PD = PE = PF   Incenter The angle bisectors of a triangle are concurrent, and their point of concurrency is the incenter Incenter Theorem If P is the incenter of ∆ABC, then PD = PE = PF 

Example 5 Use the diagram to the right. Find ST if S is the incenter of ΔMNP. 𝑎 2 + 𝑏 2 = 𝑐 2 𝑆𝑈 2 + 8 2 = 10 2 𝑆𝑈 2 =36 𝑆𝑈=6 𝑆𝑈=𝑆𝑅=𝑆𝑇 Therefore 𝑆𝑇=6

b) Find mSPU if S is the incenter of ΔMNP. 𝑚∠𝑁𝑀𝑃=31+31=62° 𝑚∠𝑀𝑁𝑃=180−56−62=62° 𝑚∠𝑆𝑃𝑈= 1 2 62 =31°

YOU TRY! In the figure shown, ND= 5 x – 1 and NE = 2 x + 11. 1. Find NF ND = NE 5𝑥−1=2𝑥+11 𝑥=4 2(4) = 11 NE = 11 ND = NE = NF NF = 11