Question

With the vertices of a $△$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, what is the sum of the radii of the three circles?

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

Answer 1

The correct option is D $9$

Let $AB=4cm$

$AC=6cm$

$BC=8cm$

Let the radius of circle 1 $\left(AP\right)$ be $x$. $AP=AQ=x$.

$\Rightarrow BP=4-x$ and $QC=6-x$

As QC and RC are radius of same circle, there $RC=QC=6-x$.

$RC+BR=8$

$\Rightarrow BR=8-(6-x)=x+2$

As $BP$ and $BR$ are radius of same circle, therefore

$4-x=x+2$

$\Rightarrow x=1$

Radius of 1st circle $=x=1cm$

Radius of 2nd circle $=4-x=3cm$

Radius of 3rd circle $=6-x=5cm$

Sum of all radii $=9cm$.