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 \( B \). The lengths of the sides are \( BC = 7 \, \text{cm} \) and \( AB = 24 \, \text{cm} \), and we need to find the radius of the inscribed circle.

Step 2: Use the formula for the radius of the incircle:
The formula for the radius \( r \) of the incircle of a right-angled triangle is given by:
\[ r = \frac{a + b - c}{2} \] where \( a \) and \( b \) are the lengths of the two perpendicular sides, and \( c \) is the length of the hypotenuse.

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

Step 4: Apply the formula for the radius:
Now, we can apply the formula for the radius \( r \), using \( a = 24 \), \( b = 7 \), and \( c = 25 \):
\[ r = \frac{24 + 7 - 25}{2} = \frac{6}{2} = 3 \, \text{cm} \]

Conclusion:
The radius of the inscribed circle is \( 3 \, \text{cm} \).