Distance Between Parallel Lines Calculator

What is the Distance Between Parallel Lines?

The distance between parallel lines is the shortest length of a line segment that is perpendicular to both lines and has its endpoints on these lines. This distance remains constant at all points along the parallel lines.

How to Calculate the Distance Between Parallel Lines

To find the distance between two parallel lines, we follow these steps:

1. Ensure the lines are in general form: \(ax + by + c = 0\)
2. Verify that the lines are indeed parallel
3. Choose a point on one line
4. Calculate the perpendicular distance from this point to the other line

Formula

The formula for the distance \(d\) between two parallel lines \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) is:

\[ d = \frac{|c_2 - c_1|}{\sqrt{a_1^2 + b_1^2}} \]

Where:

• \(a_1, b_1, c_1\) are coefficients of the first line
• \(a_2, b_2, c_2\) are coefficients of the second line
• \(|\cdot|\) denotes absolute value

Calculation Steps

1. Verify parallelism: Check if \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\)
2. Calculate \(|c_2 - c_1|\)
3. Calculate \(\sqrt{a_1^2 + b_1^2}\)
4. Divide the results from steps 2 and 3

Example and Visual Representation

Let's find the distance between these parallel lines:

\(L_1: 3x - 4y + 5 = 0\)

\(L_2: 3x - 4y - 7 = 0\)

Step 1: Verify parallelism (already confirmed)

Step 2: \(|c_2 - c_1| = |-7 - 5| = 12\)

Step 3: \(\sqrt{a_1^2 + b_1^2} = \sqrt{3^2 + (-4)^2} = 5\)

Step 4: \(d = \frac{12}{5} = 2.4\)

In this diagram, the blue line represents L₁ and the red line represents L₂. The green dashed line shows the perpendicular distance between the parallel lines, which is 2.4 units. This distance remains constant for any perpendicular line segment drawn between these parallel lines.