Question

In the above 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? 
 

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

Hint:

For a right triangle inscribed in a circle with the center as one vertex, the two sides meeting at the right angle are radii of the circle.

Answer 1

The Correct Option is D

In this problem, we are given that the area of the right triangle MON is 50 cm², and we need to find the area of the circle.
Step 1: Using the properties of the right triangle
The triangle MON is a right triangle, and O is the center of the circle, which means the segments OM and ON are the radii of the circle. Therefore, the area of triangle MON can be written as: \[ \text{Area of triangle MON} = \frac{1}{2} \times \text{base} \times \text{height} \] where the base and height are the radii of the circle, i.e., OM = ON = r.
Thus, the area of triangle MON becomes: \[ \frac{1}{2} \times r \times r = 50 \] This simplifies to: \[ \frac{1}{2} r^2 = 50 \quad \Rightarrow \quad r^2 = 100 \] Step 2: Finding the area of the circle The area of a circle is given by: \[ \text{Area of circle} = \pi r^2 \] Since \( r^2 = 100 \), we substitute this into the formula for the area of the circle: \[ \text{Area of circle} = \pi \times 100 = 100\pi \] Thus, the area of the circle is \( 100\pi \) cm². Final Answer: (D)