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A circle is inscribed in a triangle with sides of lengths 7 cm, 8 cm, and 9 cm.

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Question: A circle is inscribed in a triangle with sides of lengths 7 cm, 8 cm, and 9 cm. What is the radius of the inscribed circle?

Options:

  1. 4 cm
  2. 3 cm
  3. 2 cm
  4. 5 cm

Correct Answer: 3 cm

Solution: 

The semi-perimeter s = (7 + 8 + 9)/2 = 12 cm. The area A can be calculated using Heron\'s formula. The radius r = A/s. The area is 24 cm², so r = 24/12 = 2 cm.

A circle is inscribed in a triangle with sides of lengths 7 cm, 8 cm, and 9 cm.

Practice Questions

Q1
A circle is inscribed in a triangle with sides of lengths 7 cm, 8 cm, and 9 cm. What is the radius of the inscribed circle?
  1. 4 cm
  2. 3 cm
  3. 2 cm
  4. 5 cm

Questions & Step-by-Step Solutions

A circle is inscribed in a triangle with sides of lengths 7 cm, 8 cm, and 9 cm. What is the radius of the inscribed circle?
  • Step 1: Find the semi-perimeter of the triangle. Add the lengths of the sides: 7 cm + 8 cm + 9 cm = 24 cm. Then divide by 2: 24 cm / 2 = 12 cm. So, the semi-perimeter s = 12 cm.
  • Step 2: Use Heron's formula to find the area of the triangle. First, calculate the semi-perimeter s = 12 cm. Then, use the formula A = √(s * (s - a) * (s - b) * (s - c)), where a, b, and c are the side lengths (7 cm, 8 cm, and 9 cm). So, A = √(12 * (12 - 7) * (12 - 8) * (12 - 9)) = √(12 * 5 * 4 * 3) = √(720) = 24 cm².
  • Step 3: Calculate the radius of the inscribed circle using the formula r = A / s. We found A = 24 cm² and s = 12 cm. So, r = 24 cm² / 12 cm = 2 cm.
  • Semi-perimeter – The semi-perimeter of a triangle is half the sum of its side lengths, used in various triangle-related calculations.
  • Heron's Formula – A formula to calculate the area of a triangle when the lengths of all three sides are known.
  • Inradius – The radius of the inscribed circle of a triangle, calculated using the area and semi-perimeter.
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