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A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

1. $2 : 3$
2. $3 : 4$
3. $1 : 4$
4. $1 : 2$

Let’s draw the diagram.

Let the side of the square be $x.$ And

DIAGRAM

the radius of inscribed circle be $r$ and the radius of circumcircle be $\text{R}.$

Now, $r=\frac{x}{2}$

The Area of inscribed circle $= \pi \; r^{2} = \pi \; \left(\frac{x}{2}\right)^{2} = \frac{\pi \; x^{2}}{4}$

The diagonal of the square will be the diameter of the circumscribed circle.

So, $\text{R} = \frac{\sqrt{2} \; x}{2}$

The area of the circumscribed circle $= \pi \; \text{R}^{2} = \pi \; (\frac{\sqrt{2} \; x}{2})^{2} = \frac{\pi \; 2 x^{2}}{4} = \frac{\pi \; x^{2}}{2}$

The required ratio $= \frac{\pi \; x^{2}}{4} : \frac{\pi \; x^{2}}{2} = \frac{1}{4} : \frac{1}{2} = 1:2$

Correct Answer $: \text{D}$
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