# Given two angle and a segment, can you find $h$?

by b00n heT   Last Updated September 11, 2019 18:20 PM - source

While designing another problem I came up with the following question:

Considering the picture below, can you determine $$h$$ based only on the information of $$\alpha$$, $$\beta$$ and $$x$$ given the fact that the segment $$x$$ is the continuation of the height?

My guess is that the problem is not solvable (in the sense that one can construct multiple $$h$$ with the same $$\alpha,\beta$$ and $$x$$), but cannot see how to show this.

So far I gave the problem a try by splitting the angle and combining some trigonometric identities but couldn't conclude.

So far I only obtained the following \begin{align*} \tan(\alpha)=\frac{(h_1+h_2)(x+y)}{(x+y)^2-h_1h_2} \end{align*} and $$\tan(\beta)=\frac{(h_1+h_2)\cdot y}{y^2-h_1h_2}.$$