In the given figure. O is the center of the circle and, M and N lie on the circle. The area of the right triangle MON is 50 cm2. What is the area of the circle in cm2?

The correct option is 100π.

Let's analyze the given figure step-by-step to find the area of the circle:

Step 1: Identify the relationship between the circle and the triangle

In the provided image, we can observe a circle with center

$O$

and two points

$M$

and

$N$

lying on the circumference of the circle. The line segments

$OM$

and

$ON$

connect the center of the circle to points on the circle, which means both represent the radius of the circle, let's denote it as

$r$

Therefore, we have:

$OM=ON=r$

Step 2: Find the radius using the area of the right-angled triangle

The figure shows that the angle

$\angle MON$

is a right angle:

${90}^{\circ }$

This makes the shaded region a right-angled triangle

$MON$

where the base is

$ON$

and the height is

$OM$

The formula for the area of a right-angled triangle is:

$\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}$

Substituting base and height as the radius

$r$

we get:

$50=\frac{1}{2}\times r\times r$

Simplifying this expression:

$50=\frac{1}{2}{r}^2$

Multiplying both sides by 2:

$r^2=100$

Step 3: Calculate the area of the circle

The area of a circle is calculated using the formula:

$\text{Area of circle}=\pi r^2$

Substituting

$r^2=100$

into the formula, we find:

$\text{Area of circle}=100\pi$

Thus, the area of the circle is 100π cm².