Question

The sides of a rectangle are chosen at random, each less than a given length a, all such lengths being equally likely. The chance that the diagonal is less than $\alpha$ is

• A. $\pi /4$
• B. $\pi /2$
• C. $\pi /3$
• D. $\pi /6$

Answer 1

Answer: (A)

$\pi /4$

Let $l_1,l_2<$ a be the sides of the rectangle $\Rightarrow \sqrt{l_1^2+l_2^2}$
$=$ length of diagonal $\sqrt{2a}.$ Favourable region is the interior $\frac{1}{4}$ of a circle of radius 'a' units
$\therefore n\left(E\right)-$ square units
$n\left(S\right)=a^2$ square units
$n\left(E\right)=\frac{\pi a^2}{4}$ sq units
$\therefore P\left(E\right)=\frac{\pi }{4}$