InterviewSolution
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InterviewSolution
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If `Delta` is the area of a triangle with side lengths ` a, b, c,` then show that as `Delta leq 1/4 sqrt((a + b + c) abc)` Also, show that the equality occurs in the above inequality if and only if `a = b = c`. |
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Answer» We have to prove that `Delta le (1)/(4) sqrt((a + b + c)abc)` or `Delta le (1)/(4) sqrt(2 s abc)` or `Delta^(2) le (1)/(16) 2s abc` or `Delta^(2) le (1)/(16) 2s Delta 4R` or `rs le (1)/(2) sR` Hence, `R ge 2R` [which is always true in `Delta_(2)`] Alternative Method: In triangle, sum of two sides is greater than the third side So `a + b gt c, b + c gt a and c + a gt b` Now consider quantities `a + b -c, b + c -a, c + a -b` Using A.M. `ge` G.M. we get `((a + b -c) + (b + c -a))/(2) ge sqrt((a + b -c) (b + c -a))` or `b ge sqrt((a + b -c) (b + c -a))` Similarly we get `c ge sqrt((c +a -b) (b + c -a))` and `a ge sqrt((a + b -c) (c + a -b))` Multiplying we get `abc ge (a + b -c) (b + c -a) (c + a -b)` `rArr abc ge (2s -2a) (2s -2b) (2s -2c)` `rArr sabc ge 8s (s -a) (s-b) (s -c)` `rArr (a + b + c) abc ge 16 Delta^(2)` `rArr Delta le (1)/(4) sqrt((a + b + c) abc)` |
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