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Principle Of Mathematical Induction

Question

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In any triangle ABC, if $WiredFaculty$ show that $angle straight A space equals space 60 degree$

Solution

fraction numerator 1 over denominator straight c plus straight a end fraction plus fraction numerator 1 over denominator straight a plus straight b end fraction equals fraction numerator 3 over denominator straight a plus straight b plus straight c end fraction

(Given)

fraction numerator straight a plus straight b plus straight c plus straight a over denominator left parenthesis straight c plus straight a right parenthesis space left parenthesis straight a plus straight b right parenthesis end fraction space equals space fraction numerator 3 over denominator straight a plus straight b plus straight c end fraction

rightwards double arrow

space space left parenthesis 2 straight a plus straight b plus straight c right parenthesis space left parenthesis straight a plus straight b plus straight c right parenthesis space equals space 3 left parenthesis ac plus bc plus straight a squared plus ab right parenthesis

rightwards double arrow

2 straight a squared plus 2 ab plus 2 ac plus ab plus straight b squared plus bc plus ac plus bc plus straight c squared space equals space 3 ac plus 3 bc plus 3 straight a squared plus 3 ab

rightwards double arrow

straight b squared plus straight c squared space equals space straight a squared plus bc

rightwards double arrow

straight b squared plus straight c squared minus straight a squared space equals space bc

rightwards double arrow

fraction numerator straight b squared plus straight c squared minus straight a squared over denominator bc end fraction space equals space 1

rightwards double arrow

fraction numerator straight b squared plus straight c squared minus straight a squared over denominator 2 bc end fraction space equals space 1 half

(Dividing each side by 2)

rightwards double arrow

space space cos space straight A space equals space 1 half

Hence,

angle straight A space equals space 60 degree

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