Question

Area of triangle ABC whose sides are 24m, 40m and 32m is
        (a) $96{{m}^{2}}$
        (b) $384{{m}^{2}}$
        (c) $43{{m}^{2}}$
        (d) $192{{m}^{2}}$

Answer 1

Hint: Here, to find the area of the triangle we have to apply the formula $A=\sqrt{s(s-a)(s-b)(s-c)}$ where $s=\dfrac{a+b+c}{2}$, $a,b$ and $c$ are sides of the triangle.

Complete step-by-step solution -
Here, a triangle ABC is given whose sides are 24m, 40m and 32m.
We have to calculate the area of the triangle.
We have a formula to find the area of the triangle when all the three sides are given.
If a, b and c are the three sides of a triangle then the area of the triangle, A is given by the formula:
$A=\sqrt{s(s-a)(s-b)(s-c)}$ where $s=\dfrac{a+b+c}{2}$
Here, we have $a=24,b=40$ and $c=32$
First calculate $s$, $s$is half of the perimeter. i.e.
$\begin{align}
  & s=\dfrac{a+b+c}{2} \\
 & s=\dfrac{24+40+32}{2} \\
 & s=\dfrac{96}{2} \\
\end{align}$
By cancellation we get:
$s=48$
Now, by applying the formula we get:
$\begin{align}
  & A=\sqrt{s(s-a)(s-b)(s-c)} \\
 & A=\sqrt{48(48-24)(48-40)(48-32)} \\
 & A=\sqrt{48\times 24\times 8\times 16} \\
\end{align}$
In the next step, we have to simplify the values inside the square root to take possible values outside the square root.
Therefore, our equation becomes:
$\begin{align}
  & A=\sqrt{24\times 2\times 24\times 8\times 8\times 2} \\
 & A=\sqrt{{{24}^{2}}\times {{2}^{2}}\times {{8}^{2}}} \\
\end{align}$
The square root of ${{24}^{2}},{{2}^{2}}$ and ${{8}^{2}}$ are $24,2$ and $8$.
Therefore, our equation becomes:
$\begin{align}
  & A=24\times 2\times 8 \\
 & A=384 \\
\end{align}$
Therefore, the area of the triangle is $384{{m}^{2}}$.
Hence, the correct answer for this question is option (b).

Note: When all the three sides of the triangle are given, we can find its area by using the formula
$A=\sqrt{s(s-a)(s-b)(s-c)}$, if base and height are given we can use the formula $A=\dfrac{1}{2}bh$ to find the area of the triangle. Always read the question carefully before applying the formula, otherwise it may lead to wrong answers.