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I have a question where the teacher wants me to find the perimeter of a right triangle using the area which is equal to $1cm^2$. I've tried a few approaches but seem to be struggling and I can't even find the side lengths. is this even possible?

would love some help thanks

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    $\begingroup$ There are many right-angle triangles with area $1$. some have smallish perimeter, some have very big perimeter. Perhaps you were told more about the triangle. $\endgroup$ – André Nicolas Mar 18 '14 at 3:22
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If that is all the information you have, you can't find the perimeter. If the triangle is isosceles and the leg is $x$, the area is $\frac {x^2}2$, so $x=\sqrt 2$ and the perimeter is $2\sqrt 2+2$. If the triangle has a very long leg $L$ and a short leg of length $x$, the area is $\frac 12xL$, so $L=\frac 2x$. The perimeter is then $x+L+\sqrt{x^2+L^2}\approx 2L$ which can be as large as you like.

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Let $ l, b $ be the side lengths. Then, $$\dfrac {1}{2} lb=1 $$ So, $$l=\dfrac {2}{b} $$ The perimeter is $$ l+b+\sqrt {l^2+b^2} = \dfrac {2}{b}+b+\sqrt {\dfrac {4}{b^2}+b^2}$$ You should be able to simplify this expression.

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If the base is $b$ and the height is $c$ then the area is $bc/2$ and the perimeter is $b+c+\sqrt{b^2+c^2}$.

If $bc/2 = 1$ (sq. cm.) then $c = 2/b$, so the perimeter is $\displaystyle b+\frac 2 b + \sqrt{b^2 + \left(\frac 2 b\right)^2}$.

As $b$ goes from $0$ to $\sqrt{2}$, the perimeter goes from $\infty$ down to $2+2\sqrt{2}$, and as $b$ goes from $\sqrt{b}$ to $\infty$, the perimeter goes from $2+2\sqrt{2}$ back up to $\infty$.

So the smallest that the perimeter could be is $2+2\sqrt{2}$, but it could be anything bigger.

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