Q:

Coordinates of point that is on the end of perpendicular line

Lines AB and CD are perpendicular. Points A, B, D can have any random coordinates and we know there value. Line CD can have any random length and we know its value. How can we calculate coordinates of point C ?

A:

Suppose $A(a_1,a_2),B(b_1,b_2)$ and $D(d_1,d_2)$.
The line $(\epsilon_1)$ passing from these points has an equation of the form $Ax+By+C=0$.  
Since $DC$ (lets call her $\epsilon_2$ ) is perpendicular to $\epsilon_1$ lets distinquish the following  cases:
A. If $\epsilon_1$ has an equation of the form $x=x_0,x_0\in\mathbb{R}$.
Then $\epsilon_2$ has an equation of the form $y=y_0,y_0\in\mathbb{R}$. You can easily find $y_0$ from $D$ and then find the coordinates of $C$ from the given length.

B. If $\epsilon_1$ has an equation of the form $y=y_0,y_0\in\mathbb{R}$.
Then $\epsilon_2$ has an equation of the form $x=x_0,x_0\in\mathbb{R}$. You can easily find $x_0$ from $D$ and then find the coordinates of $C$ from the given length.

C.  If $\epsilon_1$ isn't perpendicular to any of the axes $xx',yy'$ then $\epsilon_1$ has an equation of the form $y=ax+b$ (find $a,b$).Then, $\epsilon_2$ has an equation of the form $y-d_2=\lambda(x-d_1)$ where $\lambda\cdot a=-1$. The circle $C$ with center $D(d_1,d_2)$ and radius the given length $r$ has an equation of the form $(x-d_1)^2+(y-d_2)^2=r^2$. Solving the system of equations for $C$ and $\epsilon_2$ will give you the desired coordinates. (Notice: you'll find 2 points in each case!)

