Find third vertex of a right triangle given two vertices and one side

Question

Right triangle

C _ _ _ _ _ _ B
  \        |_|
   \         |     A = (ax,ay)
    \        |     B = (bx,by)
     \       |     BC = L
      \      |     C = (?,?)
       \     |
        \    |
         \   |
          \  |
           \ |
            \|
             A

Okay so lets say you are given coordinates of points A, B and the length of BC. How do you find the coordinates of vertex C knowing it is a right triangle?

With the information I know I can get length AB(distance formula), length AC(Pythagorean), and then all the angles using law of sines but still at a blank on how to find C.

Edit: The sides are not necessarily parallel to or on any axis. The triangle may be rotated in any direction on the Cartesian plane.

Any guidance would be really helpful! I feel like this is a simple geometry problem and I have tried searching the internet but only find problems similar to this one and I can't figure it out.

Again any help would be really appreciated :) Thanks in advance!

Answer 1

Let $C(c_x,c_y)$. Then, we have two equations about $c_x,c_y$ :

$$\vec{BA}\cdot\vec{BC}=0\iff (a_x-b_x)(c_x-b_x)+(a_y-b_y)(c_y-b_y)=0$$$$BC=L\iff (c_x-b_x)^2+(c_y-b_y)^2=L^2$$ If you have $c_x\lt b_x$, then you can get $c_x,c_y$ by solving these.