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A circle is inscribed in a triangle. If the triangle has sides of lengths 7, 8,

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Question: A circle is inscribed in a triangle. If the triangle has sides of lengths 7, 8, and 9 units, what is the radius of the inscribed circle?

Options:

  1. 3 square units
  2. 4 square units
  3. 5 square units
  4. 6 square units

Correct Answer: 4 square units

Solution: 

The area of the triangle is 24 square units (using Heron\'s formula). The semi-perimeter is 12 units. The radius r = Area/semi-perimeter = 24/12 = 2 units.

A circle is inscribed in a triangle. If the triangle has sides of lengths 7, 8,

Practice Questions

Q1
A circle is inscribed in a triangle. If the triangle has sides of lengths 7, 8, and 9 units, what is the radius of the inscribed circle?
  1. 3 square units
  2. 4 square units
  3. 5 square units
  4. 6 square units

Questions & Step-by-Step Solutions

A circle is inscribed in a triangle. If the triangle has sides of lengths 7, 8, and 9 units, what is the radius of the inscribed circle?
  • Step 1: Identify the lengths of the sides of the triangle. They are 7, 8, and 9 units.
  • Step 2: Calculate the semi-perimeter of the triangle. The semi-perimeter (s) is found by adding the lengths of the sides and dividing by 2: s = (7 + 8 + 9) / 2 = 12 units.
  • Step 3: Use Heron's formula to find the area of the triangle. First, calculate the semi-perimeter (s) which we already found to be 12 units.
  • Step 4: Apply Heron's formula: Area = √(s * (s - a) * (s - b) * (s - c)), where a, b, and c are the side lengths. Here, Area = √(12 * (12 - 7) * (12 - 8) * (12 - 9)).
  • Step 5: Calculate the values inside the square root: Area = √(12 * 5 * 4 * 3).
  • Step 6: Simplify the multiplication: Area = √(720).
  • Step 7: Calculate the square root: Area = 24 square units.
  • Step 8: Now, find the radius of the inscribed circle (r) using the formula r = Area / semi-perimeter.
  • Step 9: Substitute the values: r = 24 / 12.
  • Step 10: Calculate the radius: r = 2 units.
  • Heron's Formula – A method to calculate the area of a triangle when the lengths of all three sides are known.
  • Inscribed Circle Radius – The radius of the circle that can be inscribed within a triangle, calculated using the area and semi-perimeter.
  • Semi-perimeter – Half of the perimeter of the triangle, used in the calculation of the inscribed circle's radius.
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