Finding the parallel sides of a trapezoid given all side lengths and height from base

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Suppose that we are given side lengths $a, b, c, d$ of a trapezoid. We know that two of them are parallel, and all values are different.

Moreover, we are given the height $h$ from the base (distance between two parallel lines).

The task is to find which sides are parallel.

I tried to form a triangle to use the triangle similarity, but I don't know how to proceed from there.

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1 Answer

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Assuming that the diagonal sides (I am not sure this is the standard way for calling them in English, so please edit this answer if needed) are $C$ and $D$ and the parallel sides are $A$ and $B$ with $A>B$, $$ A = \sqrt{C^2-h^2}+\sqrt{D^2-h^2} + B \tag{1}$$ holds by the Pythagorean theorem.
You may just check which permutation of $a,b,c,d$ fulfills this identity.

In $(1)$ I am actually also assuming that the angles on the major base are both acute.
Some signs have to be changed in $(1)$ is this is not the case.

In this case, for instance, we have $$ A = \color{red}{-}\sqrt{C^2-h^2}+\sqrt{D^2-h^2}+B.$$

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