GATE Mechanical 2021 Set 1 | GA | Question: 3

Question

In the above figure, $\textsf{O}$ is the center of the circle and, $\textsf{M}$ and $\textsf{N}$ lie on the circle. The area of the right triangle $\textsf{MON}$ is $50\;\text{cm}^{2}$. What is the area of the circle in $\text{cm}^{2}?$

• A. $2\pi$
• B. $50\pi$
• C. $75\pi$
• D. $100\pi$

Comments

if someone wants to go indepth then in case take MON + MON + MON + MON it will form an square inside having area is 200.

Then 200 = a^2 and a=10(2)^1/2

we know that diagonal of square = side * (2)^1/2

from this we get d = 20 and eventually r=10 and put in the area formula of square to get 100pai ans

hope it helps

Answer 1

Let the radius of the circle be $r\;\text{cm}.$

The area of triangle $ = \dfrac{1}{2} \times \text{base} \times \text{height}$

The area of the right triangle $\text{MON} = \dfrac{1}{2} \times r \times r = 50$

$\implies r^{2} = 100$
$\implies r = 10\;\text{cm}$

Now, the area of circle $ = \pi r^{2} = \pi (10)^{2} = 100\pi\;\text{cm}^{2}.$

So, the correct answer is $(D).$

Answer 2

Let the radius of the circle be $r\;\text{cm}.$

The area of triangle $ = \dfrac{1}{2} \times \text{base} \times \text{height}$

The area of the right triangle $\text{MON} = \dfrac{1}{2} \times r \times r = 50$

$\implies r^{2} = 100$
$\implies r = 10\;\text{cm}$

Now, the area of circle $ = \pi r^{2} = \pi (10)^{2} = 100\pi\;\text{cm}^{2}.$

So, the correct answer is $(D).$