# ChallengeFind the area of the shaded region

#### anemone

##### Paris la ville de l'amour
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Points $$\displaystyle P$$ and $$\displaystyle Q$$ are centers of the circles as shown below. Chord $$\displaystyle AB$$ is tangent to the circle with center $$\displaystyle P$$. Given that the line $$\displaystyle PQ$$ is parallel to chord $$\displaystyle AB$$ and $$\displaystyle AB=x$$ units, find the area of the shaded region.

#### anemone

##### Paris la ville de l'amour
Staff member
Moderator
Math Helper
Solution from my sweetheart:

Let the radius of the smaller circle be $$\displaystyle r$$ and the radius of the bigger circle be $$\displaystyle R$$.

Next, build a right-angled triangle where the base of it is half of $$\displaystyle AB$$, i.e. $$\displaystyle \dfrac{x}{2}$$.

Now, by applying the Pythagoras' theorem, we get

\displaystyle \begin{align*}\text{Area of shaded region}&=\pi(R^2-r^2)\\&=\pi\left(\dfrac{x}{2}\right)^2\\&=\dfrac{\pi x^2}{4}\end{align*}

MarkFL