Suppose we have an isosceles trapezoid whose length of the associated square is $20$ and the length of the hypotenuse of the triangles is $30$. I want to determine if it is possible to find the height of the triangles, I think no.
(Taken from http://mathworld.wolfram.com/IsoscelesTrapezoid.html.)
So in our case $a = 20$ and $c = 30$, we want to find $h$.
Suppose $x$ is the base of one of the triangles, note that $x$ can be free. Then $x^2 + h^2 = 30^2$ so simply pick two distinct such $x$ and then $h$ will not be unique.
Is this correct?
I agree. Referring to the figure in http://mathworld.wolfram.com/IsoscelesTrapezoid.html, given just the measurements $a$ and $c$ you can make $h$ be anything between $0$ and $c$ and then solve for $x$ where $x$ is the length labeled $\frac12(b - a)$ in the figure.