Question

On circle $O,$ points $C$ and $D$ are on the same side of diameter $\overline{AB},$ $\angle AOC = 30^\circ,$ and $\angle DOB = 45^\circ.$ What is the ratio of the area of the smaller sector $COD$ to the area of the circle?

$\dfrac{2}{9}$

$\dfrac{1}{4}$

$\dfrac{5}{18}$

$\dfrac{7}{24}$

$\dfrac{3}{10}$

Answer: D

Concepts:sector angle sum

Difficulty rating: 1160

Solution:

Since $\angle AOC,$ $\angle COD,$ and $\angle DOB$ fill the straight angle over diameter $\overline{AB},$ $\begin{aligned} \angle COD &= 180^\circ - 30^\circ - 45^\circ \\ &= 105^\circ. \end{aligned}$

The sector's share of the circle is $\dfrac{105^\circ}{360^\circ} = \dfrac{7}{24}.$

Thus, the correct answer is D.

2008 AMC 12B Problem 4

Answer: D

Difficulty rating: 1160

Solution:

Problem 4 in Other Years