Principle of Mathematical Induction

Question

Find $space space space straight capital delta comma space straight r space and space straight R comma$ when $straight a equals 22 comma space straight b equals 28 comma space straight c equals 36$

Solution

a = 22, b = 28, c = 36
∴    $straight s equals fraction numerator straight a plus straight b plus straight c over denominator 2 end fraction space equals space fraction numerator 22 plus 28 plus 36 over denominator 2 end fraction space equals space 43$
s - a = 43 - 22 = 21,
s - b = 43 - 28 = 15,
s - c = 43 - 36 = 7
   $straight capital delta space equals space square root of straight s left parenthesis straight s minus straight a right parenthesis space left parenthesis straight s minus straight b right parenthesis space left parenthesis straight s minus straight c right parenthesis end root space equals space square root of 43 space left parenthesis 21 right parenthesis space left parenthesis 15 right parenthesis space left parenthesis 7 right parenthesis end root$
   $equals space 21 square root of 215$ square units ...(i)
   $straight r space equals space triangle over straight s space equals space fraction numerator 21 square root of 215 over denominator 43 end fraction$ ...(ii)
$straight R space equals space fraction numerator abc over denominator 4 increment end fraction space equals space fraction numerator left parenthesis 22 right parenthesis space left parenthesis 28 right parenthesis space left parenthesis 36 right parenthesis over denominator 21 square root of 215 cross times 4 end fraction equals fraction numerator 264 over denominator square root of 215 end fraction$
$equals fraction numerator 264 over denominator square root of 215 end fraction cross times fraction numerator square root of 215 over denominator square root of 215 end fraction space equals space fraction numerator 264 square root of 215 over denominator 215 end fraction$ ...(iii)

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