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Question

In $triangle A B C comma i f c squared equals a squared plus b squared comma 2 s equals a plus b plus c$ then $4 s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis equals$

$s^{4}$
$b^{2} c^{2}$
$c^{2} a^{2}$
$a^{2} b^{2}$

hint Hint:

In this question we have to calculate the area of triangle use the heron formula and then using right angle triangle $increment equals 1 half b a s e cross times h e i g h t$ . Then equate them to find the answer.

The correct answer is: $a squared b squared$

In this question we are given $c squared equals a squared plus b squared$ , $2 s equals a plus b plus c$ and we have to find the value of $4 s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis$
Step1: Using the property of Right angle triangle
It is given that $c squared equals a squared plus b squared$ , therefore side $c$ is the hypotenuse and Sides $a comma b$ are base and height
We know that area of a right angle triangle is $increment equals 1 half b a s e cross times h e i g h t$
$increment equals 1 half a b$
Step2: Using Heron Formula
$increment equals square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root$
By squaring both sides we get,
$increment squared equals s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis$
By multiplying by $4$ both sides we get,
$4 increment squared equals 4 s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis$
=> $increment squared equals 1 fourth a squared b squared$
$4 increment squared equals 4 cross times 1 fourth a squared b squared$
=> $a squared b squared$

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