#### Question

Three circles touch each other externally. A triangle is formed when the centres of these circles are joined together. Find the radii of the circle, if the sides of the triangle formed are 6 cm, 8 cm and 9 cm

#### Solution 1

AB = 6 cm, AC = 8 cm and BC = 9 cm

Let radii of the circles having centers A, B and C be `r_1 , r_2 and r_3` respectively

`r_1 + r_3 =8`

`r_3+r_2=9`

`r_2+r_1=6`

adding

`r_1 + r_3 +r_3 +r_2 + r_2 +r_1 = 8+9+6`

`2(r_1 +r_2 +r_3) =23`

`r_1+r_2+r_3=11.5cm`

`r_1+9=11.5{(since r_2+r_3=9)`

`r_1=2.5cm`

`r_2 +6=11.5(since r_1 +r_3=6)`

`r_2 = 5.5cm`

`r_3+8=11.5(since r_2 + r_1=8)`

`r_3=3.5cm`

Hence , `r_1=2.5cm, r_2=5.5cm and r_3=3.5cm .`

#### Solution 2

AB = 6 cm, AC = 8 cm and BC = 9 cm

Let radii of the circles having centers A, B and C be `r_1 , r_2 and r_3` respectively

`r_1 + r_3 =8`

`r_3+r_2=9`

`r_2+r_1=6`

adding

`r_1 + r_3 +r_3 +r_2 + r_2 +r_1 = 8+9+6`

`2(r_1 +r_2 +r_3) =23`

`r_1+r_2+r_3=11.5cm`

`r_1+9=11.5{(since r_2+r_3=9)`

`r_1=2.5cm`

`r_2 +6=11.5(since r_1 +r_3=6)`

`r_2 = 5.5cm`

`r_3+8=11.5(since r_2 + r_1=8)`

`r_3=3.5cm`

Hence , `r_1=2.5cm, r_2=5.5cm and r_3=3.5cm .`