Right angle triangle only area given

Question

A right angle triangle has area 6 cm square. Is it possible to find the perimeter of the triangle?

Answer 1

you can't find it but you can know that it is greater than or equal to $\sqrt{24}$, if a,b,c (c hypotenuse) are the edges of the triangle, the area is equal to $\frac{ab}{2}$,so the perimeter equal to:

$$ a+b + \sqrt{(a+b)^2 - 2ab} = a+b+\sqrt{(a+b)^2 - 24}$$

so $a+b \geq \sqrt{24}$, and the perimeter $\geq \sqrt{24}$

Answer 2

$$6=0.5ab$$ $$p=a+b+\sqrt{a^2+b^2}$$

these two equtions with three variables( impossible to find it)

Answer 3

Let's try. Since $\frac{1}{2}bh=6$ then $bh=12$. Let's try to use the information that also $b^2+h^2=c^2$ where $c$ is the length of the hypotenuse. The thing we're looking for is $b+h+c$. We know we can eliminate $c$ by $\sqrt{b+h}$ and we can eliminate $h$ with $12/b$. So the perimeter has expression $b+12/b+\sqrt{b+12/b}$.

Looks like there's no way to actually find its number-value without knowing $b$, or at least one of the other edges.