Question

Consider the following three statements
Statement - 1: Heron’s formula is applied only for calculating the area of a triangle.
Statement - 2: If the four sides of a quadrilateral are given, then its area can be calculated by using the Heron’s formula.
Statement - 3: Heron’s formula is also known as Hero’s formula.
Which of the following is true?

• A. Statement 1 and 2 are true and Statement 3 is false.
• B. Statement 1 and 2 are false and Statement 3 is true.
• C. Statement 1 and 3 are true and Statement 2 is false.
• D. Statement 2 and 3 are false and Statement 1 is true.

Answer 1

The correct option is C Statement 1 and 3 are true and Statement 2 is false.

Statements 1 and 3 are true.
Heron's formula for area of a triangle $\left(A\right)$ with sides $a$, $b$ and $c$ is given by:
$A=\sqrt{s(s-a)(s-b)(s-c)}$, where ${}^{\prime }{s}^{\prime }$ is the semiperimeter.

But, statement - 2 is false.
To calculate the area of a quadrilateral by Heron’s formula we need length of the four sides as well as the length of any one of its diagonals.This is because if the length of the diagonl is not fixed, there can be many (infinitely many) quadrilaterals with the same set of lengths for the four sides. However, once any diagonal length if fixed (along with the lengths of the sides), then the quadrilateral becomes unique, that is, there can be only one quadrilateral with given side lengths and a given diagonal length.