Question

A circle is inscribed in a right-angled triangle ABC, right-angled at B. If BC = 7 cm and AB = 24 cm, find the radius of the circle

Answer 1

Step 1: Understanding the problem:
We are given a right-angled triangle $ABC$ with right angle at $B$, and the lengths of sides $BC = 7$ cm and $AB = 24$ cm. We need to find the radius of the inscribed circle in this triangle.

Step 2: Formula for the radius of the inscribed circle in a right-angled triangle:
The radius $r$ of the inscribed circle in a right-angled triangle is given by the formula: \[ r = \frac{a + b - c}{2} \] where $a$ and $b$ are the legs of the triangle, and $c$ is the hypotenuse.

Step 3: Find the length of the hypotenuse $AC$:
We can use the Pythagorean theorem to find the length of the hypotenuse $AC$: \[ AC = \sqrt{AB^2 + BC^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \, \text{cm} \]

Step 4: Apply the formula for the radius:
Now that we know $a = AB = 24$ cm, $b = BC = 7$ cm, and $c = AC = 25$ cm, we can substitute these values into the formula for the radius: \[ r = \frac{24 + 7 - 25}{2} = \frac{6}{2} = 3 \, \text{cm} \]

Conclusion:
The radius of the inscribed circle is 3 cm.