Question

If $r=$ inradius,$\Delta=$ Area of $\Delta A B C, s=$ semi-perimeter, then which of the following is true?

• A. $\Delta=r+ s$
• B. $\Delta=\frac{r}{s}$
• C. $\Delta=(r s)^{2}$
• D. $\Delta=r s$

Answer 1

Answer: (D)

Let the length of sides are $a, b$ and $c$, so $s=\frac{a +b +c}{2}$
Now, from the diagram area of triangle $A B C$ is

$\Delta =\Delta I B C+\Delta I C A+\Delta I A B$
$=\frac{1}{2} a r+\frac{1}{2} b r+\frac{1}{2} c r$
$\Delta =r\left(\frac{a+b+c}{2}\right)=r s$