# Determine Scalene Trapezoid with three sides and an angle adjacent to unknown side

Ran into this problem recently: I have a scalene trapezoid with parallel bases $$b_1$$ and $$b_2$$, and legs $$l_1$$ and $$l_2$$. Both base side lengths are known, but only one leg is known. In addition, one angle $$\alpha$$ is known, but that angle is adjacent to the unknown leg. Since the angles of a trapezoid leg are supplemental, the other angle adjacent to the unknown leg, $$\beta$$, is also known. The length of the midpoint $$m$$ can also be determined.

My question is, can the trapezoid be determined with this information? Specifically, I'm looking for the diagonals $$d_1$$ and $$d_2$$ as well as the unknown leg side length. Can the parallel bases be used to solve this in a simple(r) way?

• This problem is equivalent to finding triangle by two sides $l_1$ and $|b_2-b_1|$ and angle $\alpha$ opposite to one of them (to $l_1$). This problem has two solutions in general, because equation for second leg is quadratic (cosine rule). Commented Jun 28, 2022 at 7:20

Let the angle adjacent to the known leg be $$\beta\in (0,\pi/2)$$. Also let the angle $$\alpha$$ vary in the interval $$(0,\pi)$$. WLOG let the base adjacent to $$\alpha,\beta$$ have length $$b_1$$, and the known length be $$l_1$$ (adjacent to $$\beta$$). Dropping heights from the vertices on base $$b_2$$ to $$b_1$$ yields easily

$$h=l_1\sin\beta=l_2\sin\alpha\\ b_1=b_2+l_1\cos\beta+l_2\cos\alpha$$

Rearranging a bit we obtain

$$l_2=l_1\frac{\sin\beta}{\sin\alpha}\\ b_1-b_2=\frac{l_1}{\sin\alpha}\sin(\alpha+\beta)$$

For a given $$\alpha,b_1, b_2, l_1$$ we see that in principle we can express $$\beta, l_2$$ in terms of known quantities. By studying the behavior of the function $$f(x)=\sin(\alpha+x), x\in(0,\pi/2)$$ we see that for any $$\alpha\in (0,\pi)$$ there is an interval in which $$f$$ is not one-to-one and hence the same set of known parameters can possibly parametrize two different trapezoids.

The casework here is a little tedious- I'll add details when I have more time.

Imagine $$b_1$$, $$l_1$$, $$b_2$$, and $$l_2$$ as being long straight parts of a meccano set, with $$b_1$$ and $$b_2$$ joined to the ends of $$l_1$$ and $$l_2$$ joined to the other end of $$b_1$$. For now, put $$b_1$$ at the bottom and $$b_2$$ at the top, with $$l_1$$ the known leg on the left and $$l_2$$, the unknown length, at its correct angle on the right. Put $$l_1$$ and $$b_2$$ "flat", with $$l_1$$ lying on top of $$b_1$$ and $$b_2$$ continuing the line of $$l_1$$.

Now begin to to rotate $$l_1$$ anti-clockwise, so its end moves in an arc of a circle. Because $$b_2$$ is parallel to $$b_1$$ its right hand end must also follow a similar arc further to the right. You would need this arc to intersect the leg $$l_2$$. The arc may intersect the leg $$l_2$$ at zero, one, or two places. This means that in general this information does not determine the trapezium; there may be no trapezium matching the information, or there may be one or there may be two.