x - 7y = 9
2x - 14y + 3 = 0
translate both equations to slope intercept form.
slope intercept form is y = mx + b
m is the slope
b is the y-intercept
if the lines are parallel, they will have the same slope, and the distance between them will be the length of the line segment between them and perpendicular to both of them.
the arithmetic is messy and i'm not going to repeat the details here.
i'll just show you the steps and the results of each step.
a line perpendicular to them will have a slope that is a negative reciprocal of them.
the y intercept on the perpendicular line is not important because that line will be perpendicular to the two parallel lines throughout their length.
i therefore chose a y intercept of 0 because that makes subsequent calculations using that equation a lot easier.
slope intercept form of x - 7y = 9 becomes:
y = 1/7*x - 9/7
slope intercept form of 2x - 14y + 3 = 0 becomes:
y = 1/7*x - 9/7
since these two lines have the same slope, they are parallel to each other.
the slope of a line perpendicular to both of them will be the negative reciprocal of 1/7 which is equal to -7.
the equation of that line can use any y-intercept.
i'll use a y intercept of 0.
the equation of the line perpendicular to both of them becomes:
y = 7*x.
now you have the equation of the 2 parallel lines and the equation of the line perpendicular to them.
the equations are:
y1 = (1/7)x - 9/7
y2 = (1/7)x + 3/14
y3 = -7x
now you need to find the intersection points of that perpendicular line with each of the two parallel lines.
the intersection point of y1 and y3 will be when y1 = y3.
y1 = y3 when (1/7)x - 9/7 = -7x
the intersection point of y2 and y3 will be when y2 = y3.
y2 = y3 when (1/7)x + 3/14 = -7x
solve for x in each of those equations and then solve for y in each of those equations to get the intersection points.
y1 = y3 when x = .18 and y = -1.26
y2 = y3 when x = -.03 and y = .21
find the distance between these 2 points and you'll find the distance between the two parallel lines.
the formula for the distance between the endpoints of a line segment is d = sqrt((x1-x2)^2 + (y1-y2)^2)
whether you have (x1-x2) or (x2-x1) doesn't matter in this equation because the result will always be positive since you are squaring the difference.
same goes for (y1-y2).
d = sqrt((-.03 - .18)^2 + (-1.26 - .21)^2) which becomes:
d = sqrt((-.21)^2 + (-1.43)^2) which becomes:
d = .48492...
a graph of your equations is shown below:
the graph confirms the intersection points of the parallel lines with the perpendicular line.
