A geometry problem by A Former Brilliant Member

Geometry Level 3

The figure shown above is a semicircle with diameter of 12. The measure of arc $AC$ is $135^\circ$ . Point $D$ is the midpoint of arc $AC$ . Find the area of the shaded region. If your answer is in the form $\dfrac{a}{b} \pi$ , where $a$ and $b$ are positive coprime numbers, find $a+b$ .

The answer is 11.

1 solution

A Former Brilliant Member
Jul 10, 2017

Since $\angle DOA = \angle CBO = \dfrac{135}{2}=67.5^\circ$ , $OD \parallel BC$ .

Since $\triangle DCB$ and $\triangle CBO$ share the same base $(BC)$ and height, they have equal areas.

Therefore, the area of the shaded region is the area of the sector $CBO$ and that is $\dfrac{45}{360} \pi (6^2)=\dfrac{9}{2} \pi$

Finally,

$a+b=9+2=$ $\boxed{11}$

A geometry problem by A Former Brilliant Member

The answer is 11.

1 solution

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