Question

Three circles touch each other externally. The distance between their centers is $5cm$, $6cm$ and $7cm$. Find the radii of the circles.

Answer 1

Consider the given image wherein three circles touch each other externally.

Since the distances between the centres of these circles are $5cm,6cm$ and $7cm$ respectively, we have the following set of equations with respect to the above diagram
$x+y=5$ .......(1)
$z+y=6$ .......(2)
$x+z=7$ .......(3)
Adding (1), (2) and (3), we have
$2(x+y+z)=5+6+7=18$
$x+y+z=9$ .......(4)
Using (1) in (4), we have
$5+z=9\Rightarrow z=4$
Now using, (3)$\Rightarrow x=7-z=7-4=3$
And $\Rightarrow y=6-z=6-4=2$
Therefore, the radii of the circles are $3cm,2cm$ and $4cm$