A triangle with sides 7cm, 8cm, and 9cm is formed by connecting the centers of the three circles that are tangent to each other. Which of the following is the circumference of the largest circle?

Aug 9, 2018

$10 \pi \text{ cm.}$

Explanation:

Let ${r}_{1} , {r}_{2} \mathmr{and} {r}_{3}$ be the radii of the circles touching

each other.

If ${C}_{1} , {C}_{2} \mathmr{and} {C}_{3}$ are their respective centres, then, from

Geometry, we know that,

${C}_{1} {C}_{2} = {r}_{1} + {r}_{2} , {C}_{2} {C}_{3} = {r}_{2} + {r}_{3} , {C}_{3} {C}_{1} = {r}_{3} + {r}_{1}$.

So, by what is given, we have,

${r}_{1} + {r}_{2} = 7 , {r}_{2} + {r}_{3} = 8 \mathmr{and} {r}_{3} + {r}_{1} = 9. \ldots \ldots \ldots \left(\ast\right)$.

Adding these, $2 \left({r}_{1} + {r}_{2} + {r}_{3}\right) = 24 , \mathmr{and} , {r}_{1} + {r}_{2} + {r}_{3} = 12$.

Utilising $\left(\ast\right)$ in this last eqn., we get,

${r}_{1} = 4 , {r}_{2} = 3 \mathmr{and} {r}_{3} = 5$.

Consequently, the circumference of the largest circle, i.e.,

$\odot \left({r}_{3} , {C}_{3}\right)$ is $2 \pi {r}_{3} = 10 \pi \text{ cm.}$