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: Find the hypotenuse.
By Pythagoras' theorem:
\[ c = \sqrt{8^2 + 15^2} = \sqrt{64+225} = \sqrt{289} = 17 \text{ cm} \]
Step 2: Use the Area / semi-perimeter formula instead of the a+b-c shortcut.
\[ \text{Area} = \frac{1}{2} \times 8 \times 15 = 60 \text{ cm}^2, \qquad s = \frac{8+15+17}{2} = 20 \text{ cm} \]
Step 3: Compute the inradius.
\[ r = \frac{\text{Area}}{s} = \frac{60}{20} \]
\[ \boxed{3 \text{ cm}} \]