Question 1002047
The sum of the lengths of the two perpendicular sides of a right triangle is 30 centimeters. What are their lengths if the square of the hypotenuse is a minimum?
<pre>The square of the hypotenuse is the sum of the squares of the legs
Therefore, their sum is 30, and the hypotenuse (SUM OF THEIR SQUARES) is a minimum (you may recall that: {{{a^2 + b^2 = c^2}}})

Let longer leg be x, and shorter, y
Then we have: x + y = 30______y = - x + 30 ------- eq (i)
Also, {{{y = x^2 + y^2}}} -------- eq (ii)
{{{y = x^2 + (- x + 30)^2}}} ------- Substituting - x + 30 for y in eq (ii)
{{{y = x^2 + x^2 - 60x + 900 }}}
{{{y = 2x^2 - 60x + 900}}}

MINIMUM occurs at: x = {{{- b/(2a)}}}, or at: {{{x = - - 60/(2 * 2)}}}, or at {{{x = 60/4}}}, or at: x = 15
With MINIMUM occurring at x = 15, we get: 
y = - 15 + 30 ----------- Substituting 15 for x in eq (i)
y = 15
Thus, length of each leg is: {{{highlight_green(15)}}} cm