What are the asked coordinates in a kown right triangle?

Question

I've the following problem:

Find the coordinates $(n,m)$ and I know the values of $a,b,c$ (b is straight) and $(x,y)$. I also know that $n>x$ and $m>y$. How can I write $(n,m)$ in terms of the things I know?

I've no idea where to start

Answer 1

If the side $b$ is horizontal, then draw the perpendicular from $(n,m)$ to the $b$ side. The coordinates of the intersection are $(n,y)$. Then we can write $\sin\alpha$ and $\cos\alpha$ in the original right angle triangle and in the new right angle triangle as $$\sin\alpha=\frac cb=\frac{|m-y|}a\\\cos\alpha=\frac ab=\frac{|n-x|}a$$ By solving these equations you get $$m=y\pm\frac{ac}b\\n=x\pm\frac{a^2}b$$