1MOD5 L2

GEOMETRY

MODULE 5 LESSON 2

CIRCLES, CHORDS, DIAMETERS, AND THEIR RELATIONSHIPS

OPENING EXERCISE

Read the chart below and be prepared to use the theorems and relationships to solve exercises.

Equidistant: A point A is said to be equidistant from two different points B and C if 𝐴𝐵 = 𝐴𝐶.

2MOD5 L2

PRACTICE

1. In the figure, circle P has a radius of 10 and 𝐴𝐵 ⊥ 𝐷𝐸.’

a. If 𝐴𝐵 = 8, what is the length of 𝐴𝐶?

b. If 𝐷𝐶 = 2, what is the length of 𝐴𝐵?

a. Since 𝐴𝐵 ⊥ 𝐷𝐸, 𝐴𝐵 is bisected. Then 𝐴𝐶 = !"!= 4

b. Use the Pythagorean Thm on ∆𝐴𝐶𝑃 to find AC. Since PD is the

radius and 𝐷𝐶 = 2 ,

𝑃𝐶 = 𝑃𝐷 − 𝐷𝐶

𝑃𝐶 = 10− 2 = 8

PA also represents the radius, so 𝑃𝐴 = 10 and is the hypotenuse of ∆𝐴𝐶𝑃.

𝐴𝐶! + 𝑃𝐶! = 𝑃𝐴!

𝐴𝐶! + 8! = 10!

𝐴𝐶 = 10! − 8! = 36

𝐴𝐶 = 6

𝐴𝐵 = 2 𝐴𝐶 = 12

2. In the figure, circle A, 𝐴𝐹 = 𝐴𝐺 and 𝐵𝐶 = 22.

a. Find DE.

b. If 𝐴𝐹 ⊥ 𝐵𝐶 and 𝐴𝐹 = 5, find AB to the nearest tenth..

a. Since the chords BC and DE are equidistant from the center,

𝐵𝐶 = 𝐷𝐸 = 22

b. Since 𝐴𝐹 ⊥ 𝐵𝐶, 𝐵𝐶 is bisected. Then 𝐵𝐹 = !"!= 11

Use the Pythagorean Thm on ∆𝐴𝐹𝐵 to find AB.

𝐴𝐹! + 𝑃𝐵! = 𝐴𝐵!

𝐴𝐵 = 5! + 11! = 146

𝐴𝐵 = 12.1

3MOD5 L2

3. If 𝐴𝐵 = 𝐷𝐶, prove that ∠𝐴𝑂𝐵 ≅ ∠𝐷𝑂𝐶.

• 𝐴𝐵 = 𝐷𝐶 Given

• 𝐴𝑂 ≅ 𝐵𝑂 ≅ 𝐶𝑂 ≅ 𝐷𝑂 Radii of a circle are congruent.

• ∆𝐴𝐵𝑂 ≅ ∆𝐷𝐶𝑂 SSS

• ∠𝐴𝑂𝐵 ≅ ∠𝐷𝑂𝐶 Corresponding angles of congruent triangles

a. If ∠𝐴𝑂𝐵 = 72, what is the measure of ∠𝑂𝐷𝐶? (Note: Figure is not drawn to scale.)

𝑚∠𝐴𝑂𝐵 = 𝑚∠𝐷𝑂𝐶 = 72

∆𝐷𝐶𝑂 is an isosceles triangle, so

𝑚∠𝑂𝐷𝐶 = 𝑚∠𝑂𝐶𝐷

𝑚∠𝐷𝑂𝐶 +𝑚∠𝑂𝐷𝐶 +𝑚∠𝑂𝐶𝐷 = 180°

72°+ 2𝑚∠𝑂𝐷𝐶 = 180°

2𝑚∠𝑂𝐷𝐶 = 108°

𝑚∠𝑂𝐷𝐶 = 54°

HOMEWORK

Problem Set Module 5 Lesson 2, page 14

#1, #2, #3, #4, and #5. You must show all work and/or explain your answers using

theorems/relationships.

DUE: Monday, April 17, 2017