<s>[INST] Right triangle with equal permeter and height - how to find side lengths? The question:

Suppose there is a right triangle with sides $a$ and $b$ and
  hypotenuse $c$. Its perimeter is the same as its area, and $b = 6$.
  What are its side lengths?

I just cannot figure out how to do this! The second sentence isn't particularly helpful:
$$a + c + 6 = \frac{6a}{2}$$
$$a + c + 6 = 3a$$
$$???$$
And I can't get anywhere with the Pythagorean Theorem either:
$$a^2 + 36 = c^2$$
$$c^2 - 36 = a^2$$
$$(c + 6)(c - 6) = a^2$$
$$???$$
How do I solve this puzzle?
 [/INST] If $a=6$, then area is equal to $(6b)/2=3b$. If that is equal to the perimeter, then $a+b+c=3b$ so $a+c=2b$ so $6+c=2b$. You also know that $a^2+b^2=c^2$ so $36+b^2=c^2$.
From the $6+c=2b$ you get $c=2b-6$, and so $36+b^2=(2b-6)^2$. Expand and you get a quadratic on $b$. 
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