Question

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?

Answer 1

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 `17harr19` it will be `650`
Even `10harr26` will give a greater number: `686`
And `1harr35` 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+(36-a)^2=a^2+1296-72a+a^2`
Now we have to find the minimum of:
`2a^2-72a+1296` by setting the derivative to 0:
`4a-72=0->4a=72->a=18`