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

Question

If r= inradius,Δ= Area of Δ A B C, s= semi-perimeter, then which of the following is true? (A) Δ=r+ s (B) Δ=(r/s) (C) Δ=(r s)2 (D) Δ=r s

Tardigrade Admin · Accepted Answer

Let the length of sides are a, b and c, so s=(a +b +c/2) Now, from the diagram area of triangle A B C is <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/fc39870a330214142698806a2db746e9-.png" /> Δ =Δ I B C+Δ I C A+Δ I A B =(1/2) a r+(1/2) b r+(1/2) c r Δ =r((a+b+c/2))=r s

Q. If $r =$ inradius, $Δ =$ Area of $Δ A BC, s =$ semi-perimeter, then which of the following is true?

$Δ = r + s$

$Δ = \frac{r}{s}$

$Δ = (rs)^{2}$

$Δ = rs$

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

$Δ = Δ I BC + Δ I C A + Δ I A B$
$= \frac{1}{2} a r + \frac{1}{2} b r + \frac{1}{2} cr$
$Δ = r (\frac{a + b + c}{2}) = rs$

Q. If r= inradius,Δ= Area of ΔABC,s= semi-perimeter, then which of the following is true?

Δ=r+s

Δ=sr​

Δ=(rs)2

Δ=rs