Three circles are drawn with the vertices of a triangle as centres such that each circle touches the other two. If the sides of the triangle are 2 cm, 3 cm, and 4 cm, find the diameter of the smallest circle.

(a) 1 cm

(b) 3 cm

(c) 5 cm

(d) 4 cm.

Please give the working also..

Let ABC be the triangle and three circles are drawn such that each circle touches the other two.

Let x be the radius of circle drawn at A. y be the radius of circle drawn at C and z be the radius of the circle drawn at B.

Let the sides

AC = 2 cm

AB = 3 cm

BC = 4 cm

∴ AC = x + y = 2  ........ (1)

AB = x + z = 3  ........ (2)

BC = y + z = 4  ........ (3)

On adding (1), (2) & (3), we get 

2 (x + y + z) = 9 

⇒ x + y + z = 4.5  ...... (4)

Since x + y + z = 4.5

and x + y = 2 

⇒ z = 4.5 – 2

z = 2.5 cm

y + z = 4

 ⇒ x = 4.5 – 4

x = 0.5 cm

z + x = 3 

⇒ y = 1.5 cm

Hence the radius of smallest circle is 0.5 cm

∴ Diameter of the smallest circle is 0.5 × 2 = 1 cm