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 $50 \text{ cm}^2$. 
What is the area of the circle in $\text{cm}^2$ ?
To find the area of the circle, we need to first understand the relationship between the right triangle \( \triangle MON \) and the circle.
Given:
- \( O \) is the center of the circle.
- \( M \) and \( N \) lie on the circle.
- The area of \( \triangle MON \) is \( 50 \, \text{cm}^2 \).
Since \( \triangle MON \) is a right triangle with hypotenuse \( MN \) lying on the circle, \( O \) must be the midpoint of the hypotenuse \( MN \). According to the properties of the circle, this implies that the hypotenuse \( MN \) is the diameter of the circle.
The formula for the area of a right triangle is:
Area = \frac{1}{2} \times \text{base} \times \text{height}
In \( \triangle MON \), the base \( MO \) and height \( ON \) are both radii of the circle. Let's denote the radius as \( r \).
Thus, we have:
Area = \frac{1}{2} \times r \times r = 50
\frac{1}{2} r^2 = 50
r^2 = 100
Now, the area of the circle is given by the formula:
\text{Area of Circle} = \pi r^2
Substituting \( r^2 = 100 \):
\text{Area of Circle} = \pi \times 100 = 100\pi \, \text{cm}^2
Therefore, the area of the circle is \( 100\pi \, \text{cm}^2 \), which matches the option \( 100\pi \).
Important Questions from Mensuration and Geometry
In the given figure, $P, Q$, and $R$ are three points on a circle of radius 10 cm with $O$ as its center, $\overline{PQ} = \overline{RQ}$, and $\angle PQR = 45^\circ$. The figure is representative.
The area of the shaded region $PQRO$ is ______________ cm$^2$.
- A straight line $y = x - 1$ intersects a circle with center at $x = 1, y = 1$ and radius of magnitude 1 at two points. The length of the chord formed by this intersection is _______. (rounded off to three decimal places)
- The shell of a hollow spherical nanoparticle has a uniform thickness of 3 nanometers (nm). The outer radius of the nanoparticle is 5 nm. The ratio of the volume of the shell to the volume of the hollow core is ________
(Round off to one decimal place) - The volume of a sphere of diameter 1 unit is ______ than the volume of a cube of side 1 unit.
- A window is made up of a square portion and an equilateral triangle portion above it. The base of the triangular portion coincides with the upper side of the square. If the perimeter of the window is 6 m, the area of the window in $m^2$ is ______
