# How do you find the right triangle of maximum area if the sum of the lengths of the legs is 3?

Jun 3, 2015

We know that $x + y = 3$ and that the area $A = \frac{x \cdot y}{2}$ because of the right angle.

Let's use substitution and replace $y$ in the equation of the area :

$x + y = 3 \iff y = 3 - x$

$A = \frac{x \cdot \left(3 - x\right)}{2} = \frac{- {x}^{2} + 3 x}{2}$

The maximum area can be found by studying the sign of the derivative of the area :

$A ' \left(x\right) = \frac{- 2 x + 3}{2}$

The derivative $= 0$ when $- 2 x + 3 = 0 \iff x = \frac{3}{2}$.

Therefore, the area is maximal when $x = \frac{3}{2}$ :

$A = \frac{- {\left(\frac{3}{2}\right)}^{2} + 3 \cdot \left(\frac{3}{2}\right)}{2} = \frac{- \left(\frac{9}{4}\right) + \left(\frac{18}{4}\right)}{2} = \frac{9}{8}$.