Finding third point of right triangle

Question

I have the length of BC,and AC and the coordinates of points A and C. I'm trying to find the coordinates of B.

Answer 1

You need to find the sine and/or cosine of the angles that points A and/or C have with respect to the axes. Then find the same for the acute angles within the triangle. Geometry will then tell you the angles that BC has with respect to the axes and you can find the coordinates by multiplying the length of BC by these respective values and adding any offset from C if C is not on the origin.

Answer 2

Hint:

Convert the coordinates of $A$ to polar with the pole at $C$. Then replace the modulus ($\rho=AC$) by $BC$ and add/subtract $\dfrac\pi2$ to the argument $\theta$. Convert back to Cartesian.

Answer 3

I believe you can find the length of $AB$ (Pythagoras!). If you let the coordinates of $B$ be $(x,y),$ then since you know the other distances and coordinates, you can use the distance formula to find two equations in $x$ and $y$ (apply it to $|BA|$ and $|BC|$ in turn) that you can hopefully solve for $(x,y).$