Question

In a $△$ with sides $a,b$ and $c$, a semicircle touching the sides $AC$ and $CB$ is inscribed whose diameter lies on $AB$. Then, the radius of the semicircle is :

• A. $\frac{a}{2}$
• B. $\frac{\Delta }{s}$
• C. $\frac{\Delta }{a+b}$
• D. $\frac{2\Delta }{a+b}$

Answer 1

The correct option is D $\frac{2\Delta }{a+b}$

Let O be the center and r be the radius of the circle.
Then

Area of triangle $AOC=\frac{1}{2}\times AC\times r=\frac{1}{2}\times b\times r=\frac{br}{2}$

Area of triangle $BOC=\frac{1}{2}\times BC\times r=\frac{1}{2}\times a\times r=\frac{ar}{2}$

area$\Delta BOC$ $+$area $\Delta AOC$ =area$\Delta ABC$

$\Rightarrow (a+b)\frac{r}{2}=\sqrt{s(s-a)(s-b)(s-c)}$
$\Rightarrow r=\frac{2\sqrt{s(s-a)(s-b)(s-c)}}{a+b}=\frac{2\Delta }{a+b}$