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. \( 36\pi \) cm\(^2\)
• B. \( 49\pi \) cm\(^2\)
• C. \( 54\pi \) cm\(^2\)
• D. \( 64\pi \) cm\(^2\)

Hint:

Notice the sides form an 8-15-17 right triangle; inradius equals area over semi-perimeter.

Answer 1

The Correct Option is B

Step 1: Identify the triangle.
The sides are 15 cm, 8 cm and 17 cm.
Check \( 8^2 + 15^2 = 64 + 225 = 289 = 17^2 \), so this is a right triangle with legs 8 cm and 15 cm.

Step 2: Find the triangle's area.
With the right angle between the legs, area \( = \frac{1}{2} \times 8 \times 15 = 60 \) cm\(^2\).

Step 3: Find the semi-perimeter.
\( s = \frac{15+8+17}{2} = 20 \) cm.

Step 4: Find the inradius.
Inradius \( r = \frac{\text{Area}}{s} = \frac{60}{20} = 3 \) cm.

Step 5: Find the new radius and the circle's area.
The new radius is \( r + 4 = 7 \) cm.
Area of the circle \( = \pi (7)^2 = 49\pi \) cm\(^2\).

Final Answer:
The circle's area is \( 49\pi \) cm\(^2\). \[ \boxed{49\pi \text{ cm}^2} \]