  ### Home > CCG > Chapter 10 > Lesson 10.2.2 > Problem10-87

10-87.

Review what you know about the angles and arcs of circles below.

1. A circle is divided into nine congruent sectors. What is the measure of each central angle?

What is the total angle measure of a circle?

1. In the diagram at right, find $m\overarc { A E D }$ and $m∠C$ if $m∠B = 97º$.

How are inscribed angles related to intercepted arcs?

$m\overarc{AED} = 2(97º) = 194º$

$m\angle {C}= 0.5(194º)=97º$ 1. In $⊙C$ at right, $m∠ACB = 125º$ and $r = 8$ inches. Find $m\overarc{ A B }$ and the length of $\overarc{ A B }$. Then find the area of the smaller sector.

How does the angle measure relate to the arc measure?

Intercepted arcs are of equal measure to their corresponding central angles.

$m\overparen{AB}=m\angle {ACB}=125º$

Circumference of the Whole Circle $= 2πr = 2π(8) = 16π \; \text{inches}$

Area of Circle $= πr^2 = π(8)^2 = 64π \; \text{in}^2$

$m\overarc{AB}= \frac {125º}{360º}\cdot 16\pi = \frac{ 2000}{360} \pi ≈ 17.5 \text{ in}$

Area of Sector $=\frac{125º}{360º}\cdot64\pi=\frac{8000}{360}\pi\approx69.8\text{ in}^2$ 