# If the area of a right triangle is 15, what is its perimeter?

Apr 23, 2018

The perimeter is given as a function of side $a$ via $p = a + \frac{30}{a} + \sqrt{{a}^{2} + {30}^{2} / {a}^{2}}$ which has a minimum at $a = \sqrt{30}$ and is unbounded, no maximum.

#### Explanation:

It could be almost anything. Call the sides $a$ and $b$ and the hypotenuse $c$. Of course

${c}^{2} = {a}^{2} + {b}^{2}$

We know

$\frac{1}{2} a b = 15$

$b = \frac{30}{a}$

${c}^{2} = {a}^{2} + {\left(\frac{30}{a}\right)}^{2} = {a}^{2} + \frac{900}{a} ^ 2$

$c = \sqrt{{a}^{2} + \frac{900}{a} ^ 2}$

Call the periimeter $p$:

$p = a + b + c = a + \frac{30}{a} + \sqrt{{a}^{2} + \frac{900}{a} ^ 2}$

That's a function with a minimum at $a = \sqrt{30}$ (for positive $a$) and is unbounded, no maximum.