Geometry/triangle problem determining leg-length

Question

I have been struggling with this problem, and I think it should be possible to solve but right now I cannot find how.

Given two coordinates/points (x1,y1) and (x2,y2), the angle formed by legs d1 and d2 is 90°.

What are the lengths of d1 and d2?

Any help would be appreciated.

Answer 1

Hint:

The slope of the line passing through the segment $d_1$ is $$\tan(a).$$

Can you take it from here?

Answer 2

applying Pythagorean theorem in right triangle with legs $d_1$ & $d_2$, $$d_1^2+d_2^2=(\sqrt{(x_1-x_2)^2+(y_1-y_2)^2})^2=(x_1-x_2)^2+(y_1-y_2)^2\tag 1$$ $$\sin a=\frac{d_2}{d_1}\implies d_2=d_1\sin a$$

consider right triangle with legs $d_1$ & angle $a$. The other leg is $d_1\tan a$ & hypotenuse $d_1\sec a$

The horizontal distance between the points $(x_1, y_1)$ & $(x_2, y_2)$ is given as $$x_2-x_1=d_1\sec a+(y_1-y_2)\tan a$$ $$d_1=\frac{x_2-x_1-(y_1-y_2)\tan a}{\sec a}=\color{blue}{x_2-x_1-(y_1-y_2)\sin a}$$ Now, substitute the value of $d_1$ in (1) to find $d_2$.