Next we can label the distances so we can set up an equation. Lets call the distance between the cue ball and the side of the pool table a, the distance between the seven ball and the side of the pool table b, the distance between the two balls d, and the distance from the cue ball position and where it will hit the side of the table x. (Note that a and b are corrected distances with r subtracted from them). The last distance makes the distance from the seven ball to the side of the table d-x. Let's call the angle that the cue ball will be hit θ.
Because all angles in a triangle add up to 180, we can label the missing angle 90-θ. For further explanation click here. Also, because of the angle of reflection law, we can label the corresponding angle 90-θ, and that makes the final angle θ.
Now we can start setting up our equations. We can start by plugging in distances and angles into the tangent formula.
Tangent Formula: A/a = B/b
Our Equation: x/a = (d-x)/b
We can multiply a and b on both sides of the equation to get numbers out of the denominator of the equations.
Our Equation: x/a = (d-x)/b
Multiply both sides by a: x = a (d-x) /b
Multiply both sides by b: bx = a (d-x)
Now we can distribute the a on the left side of the equation.
Our Equation: bx = a (d-x)
After distribution: bx = ad - ax
