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distance between two parallel lines

We can see two parallel lines inred. Equations of these lines are written in slope intercept form(y=mx +c). Both the lines are parallel therefore their slopes (value of m) are equal. A, B, and C are three free points. We can move these three points. With the help of point A we can simultaneously change positions of both the parallel lines. We can change the distance between these lines with the help of point B and we can move the line segment c with the help of C. Distance between these two parallel lines is equal to the length of the line segment c and this length is written in black color
Question/s to think about. Find the value of (c_1-c_2)/√(1+m^2 ) and compare this with the distance between these two lines, what do we observe?