Question:medium

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?

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Notice the sides form an 8-15-17 right triangle; inradius equals area over semi-perimeter.
Updated On: Jul 21, 2026
  • \( 36\pi \) cm\(^2\)
  • \( 49\pi \) cm\(^2\)
  • \( 54\pi \) cm\(^2\)
  • \( 64\pi \) cm\(^2\)
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The Correct Option is B

Solution and Explanation

Step 1: Get the triangle's area with Heron's formula.
With sides 15, 8, 17 and semi-perimeter $s = 20$, area $= \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{20 \times 5 \times 12 \times 3} = \sqrt{3600} = 60$ cm$^2$.
This confirms the same area found by treating the triangle as right angled.

Step 2: Find the inradius from the area and semi-perimeter.
$r = \dfrac{\text{Area}}{s} = \dfrac{60}{20} = 3$ cm.

Step 3: Build the new radius.
New radius $= r + 4 = 3 + 4 = 7$ cm.

Step 4: Compute the circle's area.
Area $= \pi r'^2 = \pi (7)^2 = 49\pi$ cm$^2$.

Final Answer:
The required area is $49\pi$ cm$^2$, matching option (b). \[ \boxed{49\pi \text{ cm}^2} \]
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