Question 1206464
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x = missing side
s = semiperimeter = half of the perimeter
s = (a+b+c)/2
s = (5+5+x)/2
s = 5+0.5x


Heron's Formula
A = sqrt(s*(s-a)*(s-b)*(s-c))
A = sqrt((5+0.5x)*(5+0.5x-5)*(5+0.5x-5)*(5+0.5x-x))
A = sqrt((5+0.5x)*(0.5x)^2*(5-0.5x))
A = sqrt((0.5x)^2)*sqrt((5+0.5x)(5-0.5x))
A = 0.5x*sqrt(5^2 - (0.5x)^2)
A = 0.5x*sqrt(25 - 0.25x^2)


Use a graphing calculator or differential calculus to determine the max point occurs at roughly (7.071067811865, 12.5)
Note: 5*sqrt(2) = sqrt(50) = 7.071067811865 approximately


This will mean the third side being roughly 7.071067811865 units will produce a max triangle area of 12.5 square units.
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