2008 AMC 12B Problem 4

Problem 4 of 25EasierGeometry

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?

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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.