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what is the derivation of herons formula?
Asked by apurva151 | 11 Feb, 2015, 04:27: PM
answered-by-expert Expert Answer
L e t space A B C space b e space a space t r i a n g l e space w i t h space s i d e s comma space a comma space b space a n d space c W e space k n o w space t h a t space t h e space a r e a space o f space t h e space t r i a n g l e equals 1 half cross times b a s e cross times h e i g h t C o n s i d e r space t h e space f o l l o w i n g space d i a g r a m.
L e t space x plus y equals a L e t space h space b e space t h e space a l t i t u d e space o f space t h e space t r i a n g l e space A B C. T h e n space b y space a p p l y i n g space P y t h a g o r a s space T h e o r e m space i n space t h e space r i g h t space t r i a n g l e s space comma space w e space h a v e comma h squared plus x squared equals c squared space a n d space h squared plus y squared equals b squared S i n c e space x plus y equals a comma space w e space h a v e comma space y equals a minus x rightwards double arrow space y squared equals open parentheses a minus x close parentheses squared rightwards double arrow space y squared equals a squared minus 2 a x plus x squared A d d i n g space h squared space o n space b o t h space t h e space s i d e s comma space w e space h a v e comma space h squared plus space y squared equals a squared minus 2 a x plus x squared plus space h squared rightwards double arrow b squared equals a squared minus 2 a x plus c squared rightwards double arrow 2 a x equals a squared plus c squared minus b squared rightwards double arrow x equals fraction numerator a squared plus c squared minus b squared over denominator 2 a end fraction W e space h a v e comma h squared equals c squared minus x squared space space space space space equals open parentheses c plus x close parentheses open parentheses c minus x close parentheses space space space space space equals open parentheses c plus fraction numerator a squared plus c squared minus b squared over denominator 2 a end fraction close parentheses open parentheses c minus fraction numerator a squared plus c squared minus b squared over denominator 2 a end fraction close parentheses space space space space space equals open parentheses fraction numerator 2 a c plus a squared plus c squared minus b squared over denominator 2 a end fraction close parentheses open parentheses fraction numerator 2 a c minus a squared minus c squared plus b squared over denominator 2 a end fraction close parentheses space space space space space equals fraction numerator open parentheses open parentheses a plus c close parentheses squared minus b squared close parentheses open parentheses b squared minus open parentheses a minus c close parentheses squared close parentheses over denominator 4 a squared end fraction space space space space space equals fraction numerator open parentheses a plus b plus c close parentheses open parentheses a plus c minus b close parentheses open parentheses b plus a minus c close parentheses open parentheses b minus a plus c close parentheses over denominator 4 a squared end fraction space space space space space equals fraction numerator open parentheses a plus b plus c close parentheses open parentheses negative a plus b plus c close parentheses open parentheses a minus b plus c close parentheses open parentheses a plus b minus c close parentheses over denominator 4 a squared end fraction space space space space space equals fraction numerator 2 s cross times 2 open parentheses s minus a close parentheses cross times 2 open parentheses s minus b close parentheses cross times 2 open parentheses s minus c close parentheses over denominator 4 a squared end fraction comma space space space open square brackets therefore s equals fraction numerator a plus b plus c over denominator 2 end fraction space close square brackets space space space space space equals fraction numerator 4 s cross times open parentheses s minus a close parentheses cross times open parentheses s minus b close parentheses cross times open parentheses s minus c close parentheses over denominator a squared end fraction rightwards double arrow h equals fraction numerator 2 square root of s cross times open parentheses s minus a close parentheses cross times open parentheses s minus b close parentheses cross times open parentheses s minus c close parentheses end root over denominator a end fraction S i n c e space A equals 1 half cross times a cross times h rightwards double arrow A equals 1 half cross times a cross times fraction numerator 2 square root of s cross times open parentheses s minus a close parentheses cross times open parentheses s minus b close parentheses cross times open parentheses s minus c close parentheses end root over denominator a end fraction rightwards double arrow A equals square root of s cross times open parentheses s minus a close parentheses cross times open parentheses s minus b close parentheses cross times open parentheses s minus c close parentheses end root
Answered by Vimala Ramamurthy | 12 Feb, 2015, 10:43: AM

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