# Find the height of the following rectangle

We have a rectangle ABCD, and a point P on the diagonal AC. From P we see BC at an angle $$\alpha$$. Knowing $$\alpha$$, AP and AB, find BC.

Here's a probably unnecessary illustration of the problem, with known data in red:

I am lost with this, certainly, extremely simple problem. I might use the cosine law to define BC as a function of $$\alpha$$, PC and PB, but I find no way of finding those latter. I thought of finding PB constructing a parallelogram including sides PB and AP, having BC as diagonal, but I'm stuck there since I, in my poor knowledge on solving parallelograms, I would need the other diagonal of the parallelogram to find PB. On the other hand, I looked where could I transpose $$\alpha$$ as to express its trigonometric functions using known information, but didn't found anything useful.

Sorry for coming to you with high-school problems (I feel ridiculous to be asking this after three semesters of real analysis, one of complex, and one of linear algebra). I'd appreciate, in addition to a method of resolution, ressources that might help me get a comprehensive approach to planar, high-school like geometry.

1. Find $$\angle APB = 180º - \alpha$$.
2. Use the sine rule in $$\Delta APB$$ to find $$\angle PBA$$.
3. Find $$\angle PAB = 180º - \angle APB - \angle PBA$$.
then since $$\Delta CAB$$ is right-angled, $$\tan \angle PAB = \frac{BC}{AB}$$, so you now have $$BC$$.
• A shortcut: once you know $\angle PAB,$ you can write $BC = AB \tan\angle PAB$. Feb 8, 2021 at 0:21