Three circles, each with a radius of $10 \mathrm{~cm}$, are drawn tangent to each other so that their centres are all in a straight line. These circles are inscribed in a rectangle which is inscribed in another circle. The area of the largest circle is
(A) $1000 \pi$
(B) $1700 \pi$
(C) $900 \pi$
(D) $1600 \pi$
(E) $1300 \pi$

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-145.jpg?height=331&width=333&top_left_y=192&top_left_x=1357)
Three circles, each with a radius of $10 \mathrm{~cm}$, are drawn tangent to each other so that their centres are all in a straight line. These circles are inscribed in a rectangle which is inscribed in another circle. The area of the largest circle is
(A) $1000 \pi$
(B) $1700 \pi$
(C) $900 \pi$
(D) $1600 \pi$
(E) $1300 \pi$

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-270.jpg?height=398&width=393&top_left_y=262&top_left_x=1430)

## Solution

By symmetry, the centre of the large circle is the centre of the smaller middle circle. If the constructions are made as shown and with the appropriate representation of the lengths we find, $r^{2}=30^{2}+10^{2}=1000$.

Thus, $A=\pi r^{2}=\pi(1000)=1000 \pi$.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-270.jpg?height=393&width=415&top_left_y=736&top_left_x=1408)

ANSWER: (A)