How To Calculate The Distance Between Two Parallel Lines

Parallel lines are always at the same distance from each other, which might lead the astute student to wonder how a person can compute the distance between those lines. The key lies in how parallel lines, by definition, have the same slopes. Using this fact, a student can create a perpendicular line to find the points at which to determine the distance between the lines.

Step 1

Find the slope of your parallel lines. Choose either of the lines; because they share the same slope, the result will be the same. A line is in the form of y = mx + b. The variable "m" represents the line's slope. Thus, if your line is y = 2x + 3, the slope is 2.

Step 2

Create a new line in the from y = (-1/m)x. This line has a slope that is a negative reciprocal of the original line, meaning it will pass through the original line at a right angle. For example, if your line is y = 2x + 3, you have the new line as y = (-1/2)x.

Step 3

Find the point of intersection for the original line and the new line. Set the y-values of each line equal to each other. Solve for x. Then solve for y. The solution (x, y) is the intersection. For the example, setting the y-values equal yields 2x + 3 = (-1/2)x. Solving for x requires adding (1/2)x on both sides and subtracting 3 from both sides, yielding 2.5x = -3. From here, divide by 2.5 to get x = -3 / (2.5), or -1.2. Plugging this x-value into y = 2x + 3 or y = (-1/2)x results y = 0.6. Thus, the intersection is at (-1.2, 0.6).

Step 4

Repeat the previous step with the other parallel line to get an intersection point between the perpendicular line and the second parallel line.

Step 1

Find the differences between the x-values and y-values of the intersection points. For example, if your intersection points are (-6, 2) and (-4, 1), subtract the y-values first: 1 – 2 = -1. Call this Dy. Subtract the x-values second, subtracting in the same order as you used in the y-value difference calculation. Here, -4 – (-6) = 2. Call this Dx.

Step 2

Square Dy and Dx. For the example, -1^2 = 1, and 2^2 = 4.

Step 3

Add the squared values together. For the example, 1 + 4 = 5.

Step 4

Take the square root of this number, simplifying if possible. For the example, the square root of 5 can simply be left as a square root. If you want a decimal, you can actually calculate the square root of 5 to get 2.24. This is the distance between the two parallel lines.