Parallel lines have the same slope and never meet. Drawing a parallel through a point is a basic construction in geometry, engineering drawings and road design.
How to use the Parallel Line Calculator
- First line: slope m
- First line: intercept r
- Point – x₀
- Point – y₀
Results update as you type, and everything is calculated in your browser.
How it works
The slope is copied from the given line; the intercept is chosen so that the line passes through the point. The perpendicular distance between the lines follows from the intercepts.
Tips
- Sketch the points first – a quick drawing shows whether a slope should be positive or negative.
- Keep the order of the points consistent in every formula.
- Vertical lines have an undefined slope; horizontal lines have slope zero.
- Use the same units on both axes when you measure distances or angles.
Common mistakes to avoid
- Swapping x and y coordinates.
- Subtracting the coordinates in a different order in the numerator and denominator of the slope.
- Forgetting that the distance is always positive.
Inputs explained
| Field | What to enter |
|---|---|
| First line: slope m | A number |
| First line: intercept r | A number |
| Point – x₀ | A number |
| Point – y₀ | A number |
What each result means
- Parallel line intercept b = [Point – y₀] – [First line: slope m]×[Point – x₀]
- Distance between the lines = abs([Point – y₀] – [First line: slope m]×[Point – x₀] – [First line: intercept r])÷sqrt([First line: slope m]^2 + 1)
Example values
How the main result changes when First line: slope m changes and everything else stays at the example values above:
| First line: slope m | Parallel line slope a |
|---|---|
| 1 | 1 |
| 1.5 | 1.5 |
| 2 | 2 |
| 2.5 | 2.5 |
| 3 | 3 |
| 4 | 4 |
Formula
Parallel line: y = m·x + b with b = y₀ − m·x₀Distance = |b − r| ÷ √(m² + 1)Worked example
Inputs — First line: slope m: 2; First line: intercept r: 1; Point – x₀: 3; Point – y₀: -2.
Results — Parallel line slope a = 2; Parallel line intercept b = -8; Distance between the lines = 4.02492236.
Key terms
- Parallel lines
- Lines in a plane that never meet.
Frequently asked questions
How do I know two lines are parallel?
They have the same slope and different intercepts.
What is the parallel line through (3, −2) to y = 2x + 1?
y = 2x − 8.
What about vertical lines?
They are parallel when they have the form x = constant.
What if the point is on the original line?
The "parallel" line is the same line and the distance is 0.
What formula does the Parallel Line Calculator use?
Parallel line: y = m·x + b with b = y₀ − m·x₀.