Question

$ABC$ is a triangle with $AB=10$ cm, $BC=8$ cm and $AC=6$ cm .Three circles are drawn touching each other with the vertices as their centres. Find the radii of the three circles.

Answer 1

Let the radii of the circles with $A,B$ and $C$ as centres be $r_1,r_2$ and $r_3$ respectively.

According to the given information,

$AB=10cm=r_1+r_2$ … $\left(1\right)$

$BC=8cm=r_2+r_3$ … $\left(2\right)$

$CA=6cm=r_1+r_3$ … $\left(3\right)$

Adding equations $\left(1\right)$, $\left(2\right)$ and $\left(3\right)$,

$2(r_1+r_2+r_3)=24$

$r_1+r_2+r_3=12$ … $\left(4\right)$

Subtracting $\left(1\right)$ from $\left(4\right)$,

$r_3=12-10=2$

Subtracting $\left(2\right)$ from $\left(4\right)$,

$r_1=12-8=4$

Subtracting $\left(3\right)$ from $\left(4\right)$,

$r_2=12-6=6$

Thus, the radii of the three circles are $2$ cm, $4$ cm and $6$ cm.