Question

With the vertices of a $\Delta ABC$ as centers, three circles are described each touching the other two externally. If the sides of the triangle are $4$ cm, $6$ cm and $8$ cm respectively, then the sum of the radii of the three circles equals?

• A. $10$
• B. $14$
• C. $12$
• D. $9$

Answer 1

The correct option is D $9$

Consider the above figure.

Given:-

$AB=4$ cm

$AC=6$ cm

$BC=8$ cm

To find:-

$BE+AE+FC$

$AB+BC+AC=6+4+8$

$\Rightarrow AB+BC+AC=18$

$\Rightarrow$ $(AE+BE)+(BF+FC)+(AG+GC)=18$

$\Rightarrow$ $BE+AE+\left(BE\right)+FC+AE+\left(FC\right)=18$

$\because$ $AE=AG,BE=BF,FC=FG$ (radii of same circles)

$\Rightarrow$ $BE+AE+\left(BE\right)+FC+AE+\left(FC\right)=18$

$\Rightarrow$ $2BE+2AE+2FC=18$

$\Rightarrow$ $2(BE+AE+FC)=18$

$\Rightarrow$ $BE+AE+FC=9$