In the figure (ii) given below, ABC is a triangle with AB = 10 cm, BC = 8 cm and AC = 6 cm (not drawn to scale). Three circles are drawn touching each other with vertices A, B and C as their centres. Find the radii of the three circles.

Given, AB = 10 cm, BC = 8 cm, AC = 6 cm

Let the radius of the circles with centre A, B and C be x cm, y cm and z cm.

From figure,

AB = 10 cm.

⇒ x + y = 10 cm ….(i)

BC = 8 cm.

⇒ y + z = 8 cm ….(ii)

AC = 6 cm.

⇒ x + z = 6 cm …..(iii)

Adding eqn. (i), (ii) and (iii) we get,

⇒ x + y + y + z + x + z = (10 + 8 + 6) cm
⇒ 2x + 2y + 2z = 24 cm
⇒ 2(x + y + z) = 24 cm
⇒ x + y + z = 12 cm …..(iv)

Now, (iv) - (i) we get,

⇒ x + y + z - (x + y) = (12 - 10) cm
⇒ x + y + z - x - y = 2 cm
⇒ z = 2 cm.

Also, by (iv) - (ii) we get,

⇒ x + y + z - (y + z) = (12 - 8) cm
⇒ x + y + z - y - z = 4 cm
⇒ x = 4 cm

Also, by (iv) - (iii) we get,

⇒ x + y + z - (x + z) = (12 - 6) cm
⇒ x + y + z - x - z = 6 cm
⇒ y = 6 cm.

Hence, the radius of the circle with centre A, B and C are 4 cm, 6 cm and 2 cm.