Direction: The question given below is followed by three statements. Study the question and the statements. Identify which option is necessary to answer the question.
A right-angled triangle is inscribed in a given circle. What is the area of the given circle (in `cm^2`)?
I. If the base and height of the triangle in cm are both roots of the equation `x^2-23x+120=0`
II. The sum of the base and the height of the triangle is 23 cm.
III. The height of the right-angled triangle is greater than the base of the same.

Options

A. III and either only I or only II
B. All I, II and III
C. Only II and III
D. Only I

Answer

To solve the problem, we need to determine the area of a circle in which a right-angled triangle is inscribed. The area of the circle can be calculated if we know the radius, which is half of the hypotenuse of the triangle. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Triangle and Circle Relationship**:
- A right-angled triangle inscribed in a circle has its hypotenuse equal to the diameter of the circle. Therefore, if we can find the hypotenuse of the triangle, we can find the radius of the circle.

2. **Using Statement I**:
- The first statement gives us the roots of the quadratic equation \(x^2 - 23x + 120 = 0\).
- To find the roots, we can factor the equation:
\[ x^2 - 23x + 120 = (x - 15)(x - 8) = 0 \]
- This gives us the roots \(x = 15\) and \(x = 8\). These represent the base and height of the triangle.

3. **Calculating the Hypotenuse**:
- Using the Pythagorean theorem, we can find the hypotenuse (which is the diameter of the circle):
\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2 \]
\[ \text{Hypotenuse}^2 = 15^2 + 8^2 = 225 + 64 = 289 \]
\[ \text{Hypotenuse} = \sqrt{289} = 17 \text{ cm} \]

4. **Finding the Radius**:
- The radius \(R\) of the circle is half of the hypotenuse:
\[ R = \frac{17}{2} = 8.5 \text{ cm} \]

5. **Calculating the Area of the Circle**:
- The area \(A\) of the circle is given by the formula:
\[ A = \pi R^2 = \pi \left(\frac{17}{2}\right)^2 = \pi \left(\frac{289}{4}\right) = \frac{289\pi}{4} \text{ cm}^2 \]

6. **Conclusion**:
- From the first statement alone, we were able to find the area of the circle. Therefore, Statement I is sufficient to answer the question.