Challenge Find the area of the shaded region

anemone

Paris la ville de l'amour
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#1
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.

belle_circles.png
 

anemone

Paris la ville de l'amour
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#2
Solution from my sweetheart:

geometry challenge.png


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

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