Question

A point is chosen in the given diagram .What is the probability that it is in the shaded area of the part of the circle? (Given that the diameter of the circle is equal to the measure of side of the square.)

• A. $\frac{3\pi }{(4+3\pi )}$
• B. $\frac{3\pi }{(16+3\pi )}$
• C. $\frac{\pi }{(\pi +7)}$
• D. $\frac{4\pi }{(16+3\pi )}$

Answer 1

The correct option is B

$\frac{3\pi }{(16+3\pi )}$

Let 'a' be the side of the square.

Shaded area $=\frac{3}{4}$ $\times$ area of the complete circle

(Given that it is square and angle subtended by one quarter is ${90}^{\circ }$) = $\frac{3}{4}\pi {\left(\frac{a}{2}\right)}^2$

Total area = Area of the square + Area of the shaded portion
= $a^2+\frac{3}{4}\pi {\left(\frac{a}{2}\right)}^2$

Probability that the point lies in the shaded region $=\frac{Shadedarea}{Totalarea}$
$=\frac{\frac{3}{4}\pi {\left(\frac{a}{2}\right)}^2}{(a^2+\frac{3}{4}\pi {\left(\frac{a}{2}\right)}^2}$
$=\frac{3\pi }{(16+3\pi )}$