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The sides of a triangle are in the ratio of $13:14:15$ and its perimeter is $84$ centimeter. Then the area of the triangle is: -
A. $136$ square cm
B. $236$square cm
C. $336$square cm
D. $436$square cm

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Last updated date: 20th Jun 2024
Total views: 396k
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Answer
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Hint: We will use Heron's formula to solve this question.
The Heron’s formula for the area of triangle is
$A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$
Here, ‘$a$’, ‘$b$’ and ‘$c$’ are the lengths of the three sides of the triangle.
Also, in Heron’s formula ‘$s$’ stands for semi-circumference and it is given by
\[s=\dfrac{P}{2}=\dfrac{a+b+c}{2}\]
where $P$ is the perimeter of the triangle.

Complete step-by-step answer:
According to the question, the sides of the triangle are in the ratio of $13:14:15$.
Let the lengths of the three sides of the triangle be $13x$, $14x$ and $15x$.
\[\begin{align}
  & \text{Perimeter of the triangle }=\text{ }84\text{ centimeter} \\
 & \text{side }+\text{ side }+\text{ side }=\text{ }84\text{ centimeter} \\
 & \text{13}x+14x+15x=84\text{ centimeter} \\
 & 42x=84\text{ centimeter} \\
 & x=\dfrac{84}{42} \\
 & =2
\end{align}\]
Let us now find the lengths of the sides of the triangle: -
\[\begin{align}
  & \Rightarrow a=13x\ \ =\ \ 13\ \ \times \ \ 2\ \ =\ \ 26 \\
 & \Rightarrow b=14x\ \ =\ \ 14\ \ \times \ \ 2\ \ =\ \ 28 \\
 & \Rightarrow c=15x\ \ =\ \ 15\ \ \times \ \ 2\ \ =\ \ 30 \\
\end{align}\]
seo images
\[\]
Hence, the lengths of the sides of the triangle are $26$, $28$ and $30$. So, the values of $a$, $b$ and $c$ are $26$, $28$ and $30$ respectively.

Now, we will find the value of ‘$s$’ where $s$ is the semi-circumference then,
\[\begin{align}
  & s=\dfrac{\text{Premieter of the triangle}}{2} \\
 & =\dfrac{84}{2} \\
 & =42
\end{align}\]
Now substitute the value of $s$ in the heron’s equation to find the area of the triangle.\[\begin{align}
  & A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)} \\
 & =\sqrt{42\left( 42-26 \right)\left( 42-28 \right)\left( 42-30 \right)} \\
 & =\sqrt{42\left( 16 \right)\left( 14 \right)\left( 12 \right)} \\
 & =\sqrt{112896} \\
 & =336\text{ square centimeters}
\end{align}\]

So, the correct answer is “Option C”.

Note: You can find the value of $s$ from the formula
$\begin{align}
  & s=\dfrac{a+b+c}{2} \\
 & =\dfrac{26+28+30}{2} \\
 & =42
\end{align}$
From the above formula also, we get the same value for the $s$. Please remember the formula for finding the area of the triangle that is Heron's formula which is given by
$A=\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}$