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?
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?
Show Hint
- \( 2\pi \)
- \( 50\pi \)
- \( 75\pi \)
- \( 100\pi \)
The Correct Option is D
Solution and Explanation
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)
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