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Maths-

General

Easy

Question

In $triangle A B C comma a b c sin invisible function application A divided by 2 space sin invisible function application B divided by 2 space sin invisible function application C divided by 2 equals$

$Δ^{2}$
$Δ^{2} / s$
$Δ^{2 s}$
$s / Δ^{2}$

hint Hint:

In this question we will first we will use
$sin open parentheses A over 2 close parentheses equals square root of fraction numerator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator b c end fraction end root sin open parentheses B over 2 close parentheses equals square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis over denominator a c end fraction end root sin open parentheses C over 2 close parentheses equals square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis over denominator a b end fraction end root$
then we will use
$increment equals square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root$

The correct answer is: $capital delta squared divided by s$

In this question we have to find the value of $a b c sin open parentheses A over 2 close parentheses sin open parentheses B over 2 close parentheses sin open parentheses C over 2 close parentheses$
We know that
$sin open parentheses A over 2 close parentheses equals square root of fraction numerator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator b c end fraction end root sin open parentheses B over 2 close parentheses equals square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis over denominator a c end fraction end root sin open parentheses C over 2 close parentheses equals square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis over denominator a b end fraction end root$
and $increment equals square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root$
By substituting these value in the given expression
$a b c cross times square root of fraction numerator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator b c end fraction end root cross times square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis over denominator a c end fraction end root cross times square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis over denominator a b end fraction end root$
=> $a b c cross times square root of fraction numerator left parenthesis s minus a right parenthesis squared left parenthesis s minus b right parenthesis squared left parenthesis s minus c right parenthesis squared over denominator a squared b squared c squared end fraction end root$
=> $left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis$
=> $increment squared over s$

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