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Last updated date: 30th Nov 2023
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MVSAT Dec 2023

State Heron’s formula for the area of the triangle.
A.\[\sqrt {s(s + a)(s - b)(s - c)} \]
B.\[\sqrt {s(s - a)(s + b)(s - c)} \]
C.\[\sqrt {s(s - a)(s - b)(s - c)} \]
D.None of the above

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Hint: We can calculate the area of the triangle if we know the length of all the three sides with help of Heron’s Formula. To determine the area of a triangle, we don't need to know the angle measurement.
The formula is given by \[\sqrt {s(s - a)(s - b)(s - c)} \].
This formula has a lot of uses in trigonometry, including proving the law of cosines or the law of cotangents, and so on.

Complete step-by-step answer:
According to Heron, we can find the area of any given triangle, whether it is a scalene, isosceles or equilateral, by using the formula, provided the sides of the triangle.
Suppose, a triangle \[\vartriangle ABC\], whose sides are \[a,b\] and \[c\], respectively. Thus, the area of a triangle can be given as:
\[Area = \sqrt {s(s - a)(s - b)(s - c)} \]
Here \[s\]is the semi-perimeter of the triangle. It can be found out as:
\[s = \dfrac{{a + b + c}}{2}\]
To find the area of a triangle using Heron’s formula, we have to follow two steps:
The first step is to find the value of the semi-perimeter of the given triangle.
\[s = \dfrac{{a + b + c}}{2}\]
The second step is to use Heron’s formula to find the area of a triangle.
\[Area = \sqrt {s(s - a)(s - b)(s - c)} \]
Thus, Option (C) \[\sqrt {s(s - a)(s - b)(s - c)} \] is the correct answer.
So, the correct answer is “Option C”.

Note: The point to remember is that the formula has only subtraction under the square root and not addition. The formula is applicable for all types of triangle. However, few shortcuts are given below:
In case of equilateral triangle, since all sides are equal (i.e. \[a = b = c\]), we can write the formula as-
\[Area = \sqrt {s{{(s - a)}^3}} \]
In case of isosceles triangle, since two sides are equal (i.e. \[a = b\]), we can write the formula as-
\[Area = \sqrt {s{{(s - a)}^2}(s - c)} \] or
\[Area = (s - a)\sqrt {s(s - c)} \]