Math, asked by ravikumarnarhwal, 1 year ago

If a:b= c:d then prove that (a+b+c+d) (a-b-c+d) =(a+b-c-d)(a-b+c-d)


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Answers

Answered by Sanchari98
20
Here's the solution to your question.
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ravikumarnarhwal: thnx U R genius
Answered by mysticd
3

 If \: a : b = c : d \implies ad = bc\: ---(1)

 LHS = (a+b+c+d)(a-b-c+d) \\= [(a+d)+(c+d)][(a+d)-(b+c)] \\= (a+d)^{2} - (b+c)^{2}

 \boxed {\pink { (x+y)(x-y) = x^{2} - y^{2} }}

 = a^{2}+d^{2}+2ad - (b^{2}+c^{2}+2bc) \\= a^{2}+d^{2}+2ad - b^{2} - c^{2} - 2bc

/* Rearranging the terms ,we get */

= a^{2}+d^{2}- 2\orange {bc} - b^{2} - c^{2} + 2\red{ad} \\= a^{2}+d^{2}- 2\red{ad}- b^{2} - c^{2} + 2\orange{bc}

 \blue {[ From \:(1) ]}

 = (a^{2}+d^{2}- 2\red{ad}) - (b^{2} + c^{2} - 2\orange{bc})

 = (a-d)^{2} - (b-c)^{2} \\= [(a-d)+(b-c)][(a-d)-(b-c)] \\= (a-d+b-c)(a-d-b+c) \\= (a+b-c-d)(a-b+c-d)\\= RHS

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