Question

How do you find the area of the triangle ${\text{ABC}}$ given $a = 2,b = 3,c = 4$?

Answer 1

Hint: In this question we have the three sides of the triangle given therefore we will use the heron’s formula for the area of a triangle from its sides. After doing some simplification we get the required answer.

Formula used:
$Area = \sqrt {s(s - a)(s - b)(s - c)} $, where $a,b,c$ are the sides of the triangle and $s = \dfrac{{a + b + c}}{2}$.

Complete step-by-step answer:
From the question, we have the sides:
$a = 2$
$b = 3$
$c = 4$
Now to use the heron’s formula, we have to find the value of $s$, it can be found out using the formula as:
$s = \dfrac{{2 + 3 + 4}}{2}$
On adding the terms in the numerator, we get:
$s = \dfrac{9}{2}$
On dividing the terms, we get:
$s = 4.5$.
Now to find the area, we use the formula:
$area = \sqrt {s(s - a)(s - b)(s - c)} $
On substituting the values of $a,b,c$ and $s$, we get:
$area = \sqrt {4.5(4.5 - 2)(4.5 - 3)(4.5 - 4)} $
On subtracting the terms, we get:
$area = \sqrt {4.5(2.5)(1.5)(0.5)} $
On multiplying the terms, we get:
$area = \sqrt {8.3475} $
Now on taking the square root of the term by using a scientific calculator, we get:

$area = 2.9{u^2}$, which is the required solution.

Note:
It is to be noted that since there are no units of length given in the question, we take a random unit called $u$.
And since the area is a two-dimensional value, we will use the unit as ${u^2}$.
It is to be remembered that the value $s$ in the heron’s formula is the half of the perimeter of the triangle. The perimeter of the triangle is the length of the boundary of the triangle.
There are also other formulas for finding the area of the triangle which can be used based on the data given in the question. One important formula is:
$area = \dfrac{1}{2} \times b \times h$, where $b$ is the length of the base of the triangle and $h$ is the height of the triangle