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.
Finding the Points of Intersection
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.
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.
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).
Repeat the previous step with the other parallel line to get an intersection point between the perpendicular line and the second parallel line.
Calculating the Distance
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.
Square Dy and Dx. For the example, -1^2 = 1, and 2^2 = 4.
Add the squared values together. For the example, 1 + 4 = 5.
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.
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