Question

In the following figure, each circle is tangent to the other two circles, and the two shaded circles are identical to each other. Find the ratio of the shaded region to the unshaded region.

• A. $1:1$
• B. $4:5$
• C. $5:4$
• D. $4:\pi$
• E. $\pi :4$

Answer 1

The correct option is A $1:1$

Let x is the radius of the shaded circle

$\therefore$ area of the shaded circle=$\pi r^2=\pi x^2$

As according to the question both shaded circles are identical to each other.

So the area of the $2$ shaded circles $=$ $2\left(\pi r^2\right)=2\pi r^2$

The Shaded circle are same so the radius of the shaded circle is half of the larger circle.

Radius of the larger circle $=2x$

So the area of the larger circle $=$ $2\pi r^2=\pi (2x{)}^2=\pi 4x^2$

In this $4\pi x^2$area $2\pi x^2$ is shaded so the unshaded region is $2\pi x^2$

So, the ratio of shaded and unshaded region is $1:1$.