Getting 90 degree coordinate of 2 coordinates that you know

Question

I have 2 coordinates and I need to find the third with a 90 degree angle. How could I do this?

A = 50.88259382849774,6.003988087177277
B = 50.88269282423443,6.0036662220954895
C = ?

C is 90 degrees

AB = 25m

Answer 1

You could perhaps use the converse of the Pythagorean theorem. Find $C$ so that $$(AB)^2=(AC)^2+(BC)^2.$$

Alternatively, you can use the fact that $C$ lies on the line $AC$ and $BC$.

Edit

Because there are two $45^{\circ}$ angles, the sides opposite of them will have length $\frac{\sqrt{2}}{2} \cdot AB$. Say you find this length and call it $L$. Then in order to find the coordinates of $C$ you could do the following: Let $C=(x,y).$ Then, by the distance formula, \begin{align*} (50.88259382849774-x)^2&+(6.003988087177277-y)^2\\ &=(50.88269282423443-x)^2+(6.0036662220954895-y)^2\\ &=L^2 \end{align*}

Answer 2

Given three distinct points $A,B$, and $C,\ \angle ACB=90^\circ$ if and only if $C$ lies on the circle with diameter $\overline{AB}$. So, there are an infinite number of point that satisfy your initial requirements.

I'm not entirely sure what you mean by A is 45 degrees and B is 45 degrees, since $A$ and $B$ are points, not angles. If you could clarify, then I could answer your question more specifically.

Answer 3

All right angled triangles do not have 45 degrees as one of them !

The differences in x or y you gave is so small compared to 25.

xa = 50.88259382849774; ya = 6.003988087177277 xb = 50.88269282423443; yb = 6.0036662220954895

Sqrt[(xa - xb)^2 + (ya - yb)^2] is only ~ 0.000336745 not near 25.

To find C you need to give in addition inclination of CB or CA or AB to the x- or y-axis, else your problem cannot be solved.

This is because your right angled triangle has freedom to rotate all around the origin.