Question

A circle is inscribed in a right triangle ABC, right angled at B. If the lengths of the two sides containing the right angle are 8 cm and 15 cm, find the radius of the incircle.

Hint:

An alternate formula using Area and Semi-perimeter:
\[ r = \frac{\text{Area}}{\text{Semi-perimeter}(s)} \]
Area = \(\frac{1}{2} \times 8 \times 15 = 60 \text{ cm}^2\).
\(s = \frac{8 + 15 + 17}{2} = 20 \text{ cm}\).
\[ r = \frac{60}{20} = 3 \text{ cm} \]

Answer 1

Step 1: Understanding the Question:
A circle is inscribed within a right-angled triangle. We are given the lengths of the two perpendicular sides. We need to find the inradius \(r\).

Step 2: Key Formula or Approach:
For a right-angled triangle with sides \(a\), \(b\) (containing the right angle) and hypotenuse \(c\), the radius of the incircle is given by:
\[ r = \frac{a + b - c}{2} \]

Step 3: Detailed Explanation:

• Identify the given sides:
\(a = 8 \text{ cm}\)
\(b = 15 \text{ cm}\)

• Calculate the hypotenuse \(c\) using Pythagoras theorem:
\[ c = \sqrt{a^2 + b^2} \]
\[ c = \sqrt{8^2 + 15^2} \]
\[ c = \sqrt{64 + 225} = \sqrt{289} = 17 \text{ cm} \]

• Calculate the inradius \(r\) using the formula:
\[ r = \frac{a + b - c}{2} \]
\[ r = \frac{8 + 15 - 17}{2} \]
\[ r = \frac{23 - 17}{2} \]
\[ r = \frac{6}{2} = 3 \text{ cm} \]


Step 4: Final Answer:
The radius of the incircle is 3 cm.