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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 $(D).$

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 $(D).$