# The sum of the legs of a right triangle is 36 cm. For what lengths of the sides will the square of the hypotenuse be a minimum?

Jan 8, 2016

We can do this in two ways: by lateral thinking or in the robust mathematical way,

#### Explanation:

Let's do the first way , assuming both legs are 18 cm. Then the square of the hypotenuse will be ${18}^{2} + {18}^{2} = 648$
If we change this to $17 \leftrightarrow 19$ it will be $650$
Even $10 \leftrightarrow 26$ will give a greater number: $686$
And $1 \leftrightarrow 35$ will lead to $1226$

The mathematical way:
If one leg is $a$ then the other one is $36 - a$
The square of the hypotenuse is then:
${a}^{2} + {\left(36 - a\right)}^{2} = {a}^{2} + 1296 - 72 a + {a}^{2}$
Now we have to find the minimum of:
$2 {a}^{2} - 72 a + 1296$ by setting the derivative to 0:
$4 a - 72 = 0 \to 4 a = 72 \to a = 18$