Question

Two circles with centre $AandB$ have radii $8cmand1cm$ respectively. The distance $AB$ is $13cm$. A third circle with centre $C$ and radius $rcm$ touches both circles externally. If angle $ABC$ is ${90}^{\circ }$, then value of $r$ is

• A. $11$
• B. $7.57$
• C. $5.52$
• D. $2.34$

Answer 1

Answer: (B)

$7.57$

Refer Figure

Given:

$AB=13cm$

Radius of circle with center $A=8cm$

Radius of circle with center $A=1cm$

Let radius of third circle $=r$

$\angle ABC=90°$

In $△ABC$

${AB}^2+{BC}^2={AC}^2$

$AC=AD+DC=8+r$

$BC=1+r$

$AB=13$

${13}^2+{(1+r)}^2={(8+r)}^2$

$169=(8+r+1+r)\times (8+r-1-r)$

$=(2r+9)\times 7$

$2r+9=\frac{169}{7}$

$2r=\frac{169}{7}-9$

$r=\frac{106}{14}$

$r=7.57cm$