Question

The least area of a circle circumscribing any right triangle of area S is:

• A. $\pi S$
• B. $2\pi S$
• C. $\sqrt{2}\pi S$
• D. $4\pi S$

Answer 1

The correct option is A $\pi S$

let $r$ be the radius of the circle

ince, circle circumscribes right angled triangle

Therefore, diameter is the hypotenuse of the triangle ....[ prop. of semicircle]

let $a$ & $b$ be the other sides of the triangle

Therefore, $a^2+b^2={\left(2r\right)}^2$

we know that A.M$\ge$G.M

so $\frac{a^2+b^2}{2}\ge \sqrt{a^2{b}^2}=ab$

$\Rightarrow \frac{ab}{2}\le r^2$

$\Rightarrow S\le r^2$

Area of circle$=\pi r^2\ge \pi S$