On circle O, points C and D are on the same side of diameter \overlineAB, \angle AOC = 30°, and \angle DOB = 45°. What is the ratio of the area of the smaller sector COD to the area of the circle?

2008 AMC 12B Problem 4 — Solution

2008 AMC 12B Problem 4

Below is the professionally curated solution for Problem 4 of the 2008 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 12B solutions, or check the answer key.

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Concepts:sector angle sum

Difficulty rating: 1160

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?

$\dfrac{2}{9}$

$\dfrac{1}{4}$

$\dfrac{5}{18}$

$\dfrac{7}{24}$

$\dfrac{3}{10}$

Solution:

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

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

Thus, the correct answer is D.

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