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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 $\text{cm}^{2}?$

  1. $2\pi$
  2. $50\pi$
  3. $75\pi$
  4. $100\pi$
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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 $(D).$
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