Question

$ABC$ is a triangle with $AB=10\text{cm}$, $BC=8\text{cm}$, and $AC=6\text{cm}$. Three circles are drawn touching each other with the vertices of the triangle as their centres. The radius of the smallest circle is:

• A. $1\text{cm}$
• B. $2\text{cm}$
• C. $4\text{cm}$
• D. $8\text{cm}$

Answer 1

The correct option is B $2\text{cm}$

Let $AP=AQ=x,QC=RC=y,BP=BR=z$
From the data given, we have:
$x+z=10$ ... (i)
$y+z=8$ ... (ii)
$x+y=6$ ... (iii)

(i) + (ii) + (iii) $\Rightarrow 2(x+y+z)=24$
$\Rightarrow x+y+z=12$ ... (iv)

From (i) and (iv), $y=2$
From (ii) and (iv), $x=4$
From (iii) and (iv), $z=6$

Hence, the radius of the smallest circle is $2\text{cm}$