2D Coordinate - find the length of perpendicular foot on tilted system

Question

I am given values of $x_1, y_1, x_2, y_2$, $\theta_{1}$, and $\theta_{2}$.

I want to find out the value of the Unknown height.

The Unknown height is length of the perpendicular feet between Point 2 and the line extending towards the upper right side of the page.

How can I find the length of the perpendicular feet?

Answer 1

Calculate the distance between points $P_1$ and $P_2$:

$$ d_{12} = [(x_2 - x_1)^2 + (y_2 - y_1)^2]^{1/2} $$

and note that

$$ \sin \theta_1 = \frac{\mbox{unknown height}}{d_{12}} $$

So that

$$ \mbox{unknown height} = d_{12} \sin \theta_1 $$