Question

The dimensions of a triangle are 15 cm, 8 cm and 17 cm. What is the area of a circle having radius (r + 4) cm if 'r' is the inradius of the given triangle?

• A. 36π cm²
• B. 49π cm²
• C. 54π cm²
• D. 64π cm²
• E. 81π cm²

Answer 1

The Correct Option is B

To solve this problem, we must first find the inradius of the triangle with sides 15 cm, 8 cm, and 17 cm, and then use this value to calculate the area of the circle with radius \((r + 4)\) cm.

1. First, verify whether the triangle is a right triangle. According to the Pythagorean theorem, if a triangle with sides \(a\), \(b\), and \(c\) (where \(c\) is the hypotenuse) satisfies \(a^2 + b^2 = c^2\), then it is a right triangle.
   • Calculate: \(15^2 + 8^2 = 225 + 64 = 289 = 17^2\). The triangle is a right triangle.
2. For a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), the area \(A\) is given by: \(A = \frac{1}{2} \times a \times b = \frac{1}{2} \times 15 \times 8 = 60 \text{ cm}^2\).
3. For a right triangle, the inradius \(r\) can be found using: \(r = \frac{a + b - c}{2}\). Substitute the values: \(r = \frac{15 + 8 - 17}{2} = \frac{6}{2} = 3 \text{ cm}\).
4. Now, calculate the area of the circle with radius \((r + 4) = 3 + 4 = 7 \text{ cm}\). The area of the circle is: \(\pi (r + 4)^2 = \pi \times 7^2 = 49\pi \text{ cm}^2\).
5. Therefore, the area of the circle is \(49\pi \text{ cm}^2\), which is the correct option.