Question

All the vertices of a right-angled triangle lie on the circumference of a circle. The length of the two perpendicular sides of the triangle are 14 cm and 48 cm. What is the area of the circle?

• A. \(2304 \pi \)cm²
• B. \(576 \pi \)cm²
• C. \(625 \pi \)cm²
• D. \(2500 \pi \)cm²

Answer 1

The Correct Option is C

Area of Circle Calculation 

- Given that the vertices of the right-angled triangle lie on the circumference of a circle, the hypotenuse of the triangle is the diameter of the circle.

- The sides of the right-angled triangle are 14 cm and 48 cm.

- Using the Pythagorean theorem to find the hypotenuse:

\[ h = \sqrt{14^2 + 48^2} = \sqrt{196 + 2304} = \sqrt{2500} = 50 \text{ cm} \]

- Thus, the diameter of the circle is 50 cm.

- The radius \(r\) of the circle is half of the diameter:

\[ r = \frac{50}{2} = 25 \text{ cm} \]

- The area \(A\) of the circle is given by the formula:

\[ A = \pi r^2 = \pi \times 25^2 = \pi \times 625 = 625\pi \text{ cm²} \]

Thus, the area of the circle is 625π cm².

Conclusion: The correct answer is 625π cm².