Question

Fill in the blanks choosing the appropriate option.

According to Heron's formula if ' $\mathrm{s}$ ' is the $\left(\mathrm{P}\right)$ of a triangle and $a,b,c$ are the three sides of the triangle respectively, then the area of the triangle is $\left(\mathrm{Q}\right)$

• A. $\left(\mathrm{P}\right)$ - perimeter,
  $\left(Q\right)=\sqrt{s(s-a)\left(s-b\left)\right(s-c)\right.}$
• B. $\left(\mathrm{P}\right)$-semi-perimeter,$\left(Q\right)={\left[\frac{1}{s(s-a)(s-b)(s-c)}\right]}^{-\frac{1}{2}}$
• C. $\left(\mathrm{P}\right)$ - semi-perimeter,
  $\left(Q\right)=\left[s(s-a){\left(s-b\left)\right(s-c)\right]}^{-1/2}\right.$
• D. $\left(\mathrm{P}\right)$ - perimeter,
  $\left(Q\right)=\left[s(s-a){\left(s-b\left)\right(s-c)\right]}^{-1/2}\right.$

Answer 1

The correct option is B

$\left(\mathrm{P}\right)$-semi-perimeter,

$\left(Q\right)={\left[\frac{1}{s(s-a)(s-b)(s-c)}\right]}^{-\frac{1}{2}}$

Step 1 Given data:
Semi-perimeter of triangle$=P$
Three sides of the triangle are $a,b,c$
Area of triangle$=Q$

Step 2 Heron's formula to find the area of triangle:
We know that,
If three sides $a,b,c$ of the triangle are given then area of triangle is given by,

Area of triangle$=\sqrt{s(s-a)\left(s-b\left)\right(s-c)\right.}\left(Heron\text{'}sformula\right)$
Where, semi-perimeter $\left(s\right)=\frac{a+b+c}{2}$

Step 3 Area of triangle:
We know that, $s$ is a semi-perimeter of a triangle with sides $a,b,c$ then, area of triangle $Q$ is given as,

$$\begin{gathered} Q=\sqrt{s(s-a)\left(s-b\left)\right(s-c)\right.} \\ ={\left(s(s-a)\left(s-b\left)\right(s-c)\right.\right)}^{\frac{1}{2}} \\ ={\left[\frac{1}{\left(s(s-a)\left(s-b\left)\right(s-c)\right.\right)}\right]}^{-\frac{1}{2}} \end{gathered}$$
Thus, $P$ is the semi-perimeter and $Q$ is given as ${\left[\frac{1}{\left(s(s-a)\left(s-b\left)\right(s-c)\right.\right)}\right]}^{-\frac{1}{2}}$

Hence, option $\left(b\right)$ is correct answer.