geometry module 5 lesson 2 circles, chords, … · geometry module 5 lesson 2 circles, chords,...
TRANSCRIPT
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