Two circles of radius 10 are tangent to each other. A tangent is drawn from the centre of one of the circles to the second circle. To the nearest integer, what is the area of the shaded region?
(A) 6
(D) 9
(B) 7
(C) 8
(E) 10

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-140.jpg?height=314&width=553&top_left_y=965&top_left_x=1233)
Two circles of radius 10 are tangent to each other. A tangent is drawn from the centre of one of the circles to the second circle. To the nearest integer, what is the area of the shaded region?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-282.jpg?height=377&width=658&top_left_y=511&top_left_x=1254)

## Solution

Let the centres of the circles be $B$ and $C$ and the point of contact of the tangent line from $B$ to the circle with centre $C$ be $A$. Thus, $\angle C A B=90^{\circ}$ using properties of tangents to a circle.

Since the sides of right-angled triangle $A B C$ are in the ratio of 1:2: $\sqrt{3}, \angle B=30^{\circ}$ and $\angle C=60^{\circ}$. The area of the shaded region is equal to the area of $\triangle A B C$ minus the area

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-282.jpg?height=379&width=659&top_left_y=1003&top_left_x=1275)
of the two sectors of the circles.

The area of the two sectors is equivalent to $\frac{1}{4}$ the area of a circle with radius 10 .

The area of the shaded region $=\frac{1}{2}(10)(10 \sqrt{3})-\frac{1}{4} \pi(10)^{2}$

$$
=50 \sqrt{3}-25 \pi
$$

$$
\approx 8.063
$$

To the nearest integer, the area is 8 .

ANSWER: (C)