SOLUTION: 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?

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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?
Found 3 solutions by addingup, josgarithmetic, MathTherapy:
Answer by addingup(3677)   (Show Source): You can put this solution on YOUR website!
if the square of the hypotenuse is a minimum of what??
Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
h, hypotenuse
x and 30-x are lengths of the legs.




Find the x value or values for the minimum .


-
Discriminant,
This method, finding the roots, will not work here because as a function does not intersect the x-axis.

You can complete the square for making into standard form, and then read the vertex (a minimum point) from that form of the equation. The term to use is , or 225.



Showing x=15 for the vertex, and then one leg of the triangle is 15, and therefore the other leg is also.
Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
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?
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: )



Let longer leg be x, and shorter, y

Then we have: x + y = 30______y = - x + 30 ------- eq (i)

Also,  -------- eq (ii)

 ------- Substituting - x + 30 for y in eq (ii)







MINIMUM occurs at: x = , or at: , or at , 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:  cm  

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