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Question

$n space triangle A B C space left parenthesis c o t invisible function application A divided by 2 plus c o t invisible function application B divided by 2 plus c o t invisible function application C divided by 2 right parenthesis divided by left parenthesis c o t invisible function application A plus c o t invisible function application B plus c o t invisible function application C right parenthesis equals$

$((a + b + c)^{2}) / (a^{2} + b^{2} + c^{2})$
$(a^{2} + b^{2} + c^{2}) / ((a + b + c)^{2})$
s
Δ

hint Hint:

In this question we have to use these formulas and then just simplify.
$cot open parentheses A over 2 close parentheses equals fraction numerator s left parenthesis s minus a right parenthesis over denominator increment end fraction$
$cot open parentheses B over 2 close parentheses equals fraction numerator s left parenthesis s minus b right parenthesis over denominator increment end fraction$
$cot open parentheses C over 2 close parentheses equals fraction numerator s left parenthesis s minus c right parenthesis over denominator increment end fraction$
$cot open parentheses A close parentheses equals fraction numerator b squared plus c squared minus a squared over denominator 4 increment end fraction$

The correct answer is: $left parenthesis left parenthesis a plus b plus c right parenthesis squared right parenthesis divided by left parenthesis a squared plus b squared plus c squared right parenthesis$

In this question we have to find the value of $fraction numerator cot open parentheses begin display style A over 2 end style close parentheses plus cot open parentheses begin display style B over 2 end style close parentheses plus cot open parentheses begin display style C over 2 end style close parentheses over denominator cot open parentheses A close parentheses plus cot open parentheses B close parentheses plus cot open parentheses C close parentheses end fraction$
Step1: Using the formula of the solution of triangle
$cot open parentheses A over 2 close parentheses equals fraction numerator s left parenthesis s minus a right parenthesis over denominator increment end fraction$
$cot open parentheses B over 2 close parentheses equals fraction numerator s left parenthesis s minus b right parenthesis over denominator increment end fraction$
$cot open parentheses C over 2 close parentheses equals fraction numerator s left parenthesis s minus c right parenthesis over denominator increment end fraction$
$cot open parentheses A close parentheses equals fraction numerator b squared plus c squared minus a squared over denominator 4 increment end fraction$
by putting these values in the above expression we get,
$fraction numerator begin display style fraction numerator s left parenthesis s minus a right parenthesis over denominator increment end fraction end style plus begin display style fraction numerator s left parenthesis s minus b right parenthesis over denominator increment end fraction end style plus begin display style fraction numerator s left parenthesis s minus c right parenthesis over denominator increment end fraction end style over denominator begin display style fraction numerator b squared plus c squared minus a squared over denominator 4 increment end fraction end style plus begin display style fraction numerator a squared plus c squared minus b squared over denominator 4 increment end fraction end style plus begin display style fraction numerator a squared plus b squared minus c squared over denominator 4 increment end fraction end style end fraction$
=> $fraction numerator begin display style fraction numerator s left parenthesis s minus a plus s minus b plus s minus c right parenthesis over denominator increment end fraction end style over denominator begin display style fraction numerator b squared plus c squared minus a squared plus a squared plus c squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction end style end fraction$
=> $fraction numerator begin display style fraction numerator s left parenthesis 3 s minus a minus b minus c right parenthesis over denominator increment end fraction end style over denominator begin display style fraction numerator b squared plus c squared plus a squared over denominator 4 increment end fraction end style end fraction$
We know that $a plus b plus c space equals space 2 s$
=> $fraction numerator begin display style fraction numerator s left parenthesis 3 s minus 2 s right parenthesis over denominator increment end fraction end style over denominator begin display style fraction numerator b squared plus c squared plus a squared over denominator 4 increment end fraction end style end fraction$
=> $fraction numerator begin display style 4 s squared end style over denominator begin display style b squared plus c squared plus a squared end style end fraction$
=> $fraction numerator begin display style 4 left parenthesis fraction numerator a plus b plus c over denominator 2 end fraction right parenthesis squared end style over denominator begin display style b squared plus c squared plus a squared end style end fraction$
=> $fraction numerator begin display style left parenthesis a plus b plus c right parenthesis squared end style over denominator begin display style b squared plus c squared plus a squared end style end fraction$

Related Questions to study

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parallel

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