Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively and $\angle B=\frac{\pi }{2}$, then value of radius of incircle is?

• A. $\frac{a+b-c}{2}$
• B. $\frac{a-b+c}{2}$
• C. $\frac{a+b+c}{2}$
• D. none of these

Answer 1

The correct option is A $\frac{a+b-c}{2}$

Considering the image we get
$a+b-c$
$=AC+BC-AB$
$=(AD+CD)+(CE+BE)-(AF+BF)$
Now
$AD=AF$, $CD=CE$ and $BE=BF$
Hence simplifying, we get
$=2CD$
Now
$CDOE$ is a square.
Hence $2CD=2r_{incircle}$
Or
$r_{incircle}=\frac{a+b-c}{2}$.