ABC is a right-angled triangle right angled at B. A circle is inscribed in it in the lengths of the two sides containing the right angle are 5 cm and 12 cm. Find the radius of the circle.

Given: AB = 12 cm

BC = 5 cm

To find: the value of r

Theorem Used:

Pythagoras theorem:

In a right-angled triangle, the squares of the hypotenuse is equal to the sum of the squares of the other two sides.

Explanation:

Using the Pythagoras theorem stated above,

AC² = AB² + BC²

⇒ AC² = 12² + 5²

⇒ AC² = 144 + 25

⇒ AC² = 169

⇒ AC = √169

⇒ AC = 13 cm

Now,

Area of ΔABC = Area of ΔOAB + Area of ΔOBC + Area of ΔOCA

As we know,

Area of triangle = 1/2 × base × height

⇒ 1/2 × BC × AB = (1/2 × AB × r) + (1/2 × BC × r) + (1/2 × AC × r)

⇒ 1/2 × 5 × 12 = (1/2 × 12 × r) + (1/2 × 5 × r) + (1/2 × 13 × r)

⇒ 30 = 6r + 2.5r + 6.5r

⇒ 30 = 6r + 2.5r + 6.5r

⇒ 30 = 15r

⇒ r = 2 cm