Question

A triangle has sides with lengths: 7, 2, and 14. How do you find the area of the triangle using Heron’s formula?

Answer 1

Hint: We are given a triangle with sides of length 7, 2 and 14.
We are asked to find the area of the triangle using Heron’s formula. To do so, we will learn about the Heron's formula, How are area calculated, what is semi perimeter, and perimeter, we will also learn what condition should be satisfied by the triangle to get exist.

Complete step-by-step solution:
We are given a triangle with sides of length as 7, 2 and 14, we will name them as a, b, and c.
So, we have $a=7,b=2\text{ and }c=14$ .
Now, we will learn Heron’s formula.
Before we start, we will learn about perimeter and semi perimeter of any figure is refer as the sum of all sides and semi perimeter area refer as half of the perimeter, it is denoted by ‘S’ and given as –
$S=\dfrac{a+b+c}{2}$ .
Now in any triangle with sides say a, b, c.
The Heron's formula says that the area of the triangle is given as –
$\text{Area}=\sqrt{S\left( S-a \right)\left( S-b \right)\left( S-c \right)}$
Where, S=semi perimeter.
a, b, c are the sides of the triangle.
Remember that semi perimeter should always be greater than individual sides.
Now, we work on our problem.
We have sides as 7, 2, 14.
So, considered $a=7,b=2\text{ and }c=14$ .
So, we will find a semi perimeter.
$S=\dfrac{a+b+c}{2}$ .
As $a=7,b=2\text{ and }c=14$ .
So, $S=\dfrac{7+2+14}{2}$
$\begin{align}
  & =\dfrac{23}{2} \\
 & S=11.5 \\
\end{align}$
So, we get semi perimeter as $S=11.5$ which is less than the third side.
So, we cannot find the area as the semi perimeter should always be greater than all the individual sides.
So, for our triangle with sides’ length 7, 2 and 14, no such triangle exists.

Note: Triangles exist if the sum of any two sides is always greater than remaining sides.
Say if we have sides a, b and c.
Then it should hold true.
$\begin{align}
  & a+b>c \\
 & a+c>b \\
 & a+b+c>a \\
\end{align}$
In our triangle we have sides as 7, 2, 14.
So, consider $a=7,b=2\text{ and }c=14$ .
So, here $a+b=7+2=9$ which is not greater than $c=14$ .
So, here the sum of 7 and 2 is not greater than the third sides of length 14cm.
Hence, such triangles do not exist.