Question

If the area of a triangle is 81 square cm and its perimeter is 27 cm, then its in-radius is
A. 6 cm
B. 3 cm
C. 1.5 cm
D. none of these

Answer 1

Hint: The general formula for the in-radius of a triangle is given by $\dfrac{A}{s}$, where A is the area of triangle and s is the semi-perimeter of the triangle. So by substituting the values and simplifying the obtained equation we will get the desired answer.

Complete step by step answer:
We have been given that the area of a triangle is 81 square cm and its perimeter is 27 cm.
We have to find the in-radius of a triangle.
Now, we know that the in-radius of a triangle is the radius of the largest circle that will fit inside the triangle. The in-radius of a triangle is given by $r=\dfrac{A}{s}$, where A is the area of the triangle and s is the semi-perimeter of the triangle.
Now, the semi-perimeter of the triangle is half the perimeter of a triangle. We have given the perimeter of the triangle 27 cm, so the semi-perimeter of the triangle will be
$\Rightarrow s=\dfrac{27}{2}$
We have the area of triangle $A=81c{{m}^{2}}$
Now, substituting the values in the formula of in-radius we will get
$\Rightarrow r=\dfrac{81}{\dfrac{27}{2}}$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow r=81\times \dfrac{2}{27} \\
 & \Rightarrow r=3\times 2 \\
 & \Rightarrow r=6cm \\
\end{align}$

Hence we get the in-radius of a triangle as 6 cm.

Note: Alternative formula to find the in-radius of a triangle is given by $r=\dfrac{\sqrt{\left( s-a \right)\left( s-b \right)\left( s-c \right)}}{s}$, where s is the semi-perimeter of a triangle and a, b, c are the sides of a triangle. The point to be remembered is that in-radius of a triangle inscribed inside the triangle.