GEOMETRYCoordinate GeometryMathematics Calculator
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Distance Between Parallel Lines

For parallel lines Ax+By+C₁=0 and Ax+By+C₂=0 (same A,B), the perpendicular distance is d = |C₁−C₂|/√(A²+B²). Pick any point on one line and use point-to-line distance.

Concept Fundamentals
|C₁−C₂|/√(A²+B²)
d
Same A, B (parallel)
Condition
Shortest distance
Perpendicular
C₁=C₂ ⇒ d=0
Coincident

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Same formula as point-to-line distance from (0,−C₁/B) to second line. d = 0 when lines coincide (C₁=C₂). Distance is independent of which line is "first".

Key quantities
|C₁−C₂|/√(A²+B²)
d
Key relation
Same A, B (parallel)
Condition
Key relation
Shortest distance
Perpendicular
Key relation
C₁=C₂ ⇒ d=0
Coincident
Key relation

Ready to run the numbers?

Why: Distance between parallel lines appears in geometry, tolerance analysis, and path planning. The formula is the same as point-to-line distance from any point on one line to the other.

How: d = |C₁−C₂|/√(A²+B²). Lines must be parallel (same A and B). If C₁=C₂, lines coincide and d=0. The distance is measured perpendicular to both lines.

Same formula as point-to-line distance from (0,−C₁/B) to second line.d = 0 when lines coincide (C₁=C₂).
Sources:Khan Academy — DistanceWolfram MathWorld — Line-Line Distance

Run the calculator when you are ready.

Find DistanceEnter parallel lines Ax+By+C₁=0, Ax+By+C₂=0

Parallel Lines (Ax + By + C = 0)

Both lines share the same A and B. Only C differs.

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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d = |C₁−C₂|/√(A²+B²) for parallel lines.

— Coordinate Geometry

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Perpendicular distance is the shortest.

— Property

Key Takeaways

  • • Parallel lines have the same slope (same A and B in Ax+By+C=0)
  • • Formula: d=frac∣C1−C2∣sqrtA2+B2d = \\frac{|C_1 - C_2|}{\\sqrt{A^2 + B^2}}
  • • The distance is measured perpendicular to both lines
  • • For horizontal lines y=k₁ and y=k₂: d = |k₁−k₂|
  • • For vertical lines x=k₁ and x=k₂: d = |k₁−k₂|

Did You Know?

Constant Distance

The distance between parallel lines is constant — the same at every point along the lines.

Derivation

Pick any point on L₁, find its perpendicular distance to L₂. Same as point-to-line distance with |C₁−C₂|.

Road Lanes

Lane width on roads is the distance between parallel lane markings. Standard is about 3.5–4 meters.

Railway Tracks

Rail gauge (distance between rails) is measured as the perpendicular distance between the inner edges.

Architecture

Used for spacing of parallel walls, beams, and structural elements.

Coordinate Geometry

If lines are not parallel, they intersect and the distance is 0 at the intersection.

Understanding the Formula

For parallel lines Ax+By+C1=0Ax + By + C_1 = 0 and Ax+By+C2=0Ax + By + C_2 = 0:

d=frac∣C1−C2∣sqrtA2+B2d = \\frac{|C_1 - C_2|}{\\sqrt{A^2 + B^2}}

The numerator is the difference in intercepts (scaled). The denominator normalizes by the length of the normal vector (A, B).

Expert Tips

Same A and B

Both lines must have identical A and B. Only C differs. If A,B differ, lines are not parallel.

Horizontal Lines

For y=k₁ and y=k₂, use A=0, B=1, C₁=−k₁, C₂=−k₂. Distance = |k₁−k₂|.

Vertical Lines

For x=k₁ and x=k₂, use A=1, B=0, C₁=−k₁, C₂=−k₂. Distance = |k₁−k₂|.

Coincident

If C₁=C₂, the lines are the same and distance is 0.

Frequently Asked Questions

What if the lines are not parallel?

This formula applies only to parallel lines. Non-parallel lines intersect, so the distance between them is 0 at the intersection.

Why must A and B be the same?

Parallel lines have the same direction. In Ax+By+C=0, (A,B) is the normal vector. Same normal ⇒ same direction ⇒ parallel.

Can the distance be zero?

Yes, when C₁=C₂. Then the lines are coincident (the same line).

How do I convert y=mx+b?

For y=mx+b₁ and y=mx+b₂, rewrite as mx−y+b₁=0 and mx−y+b₂=0. A=m, B=−1, C₁=b₁, C₂=b₂.

What about vertical lines?

For x=k₁ and x=k₂, use A=1, B=0, C₁=−k₁, C₂=−k₂. Distance = |k₁−k₂|.

Units of distance?

Same units as the coordinate system. If x,y are in meters, distance is in meters.

Real-world applications?

Road lane width, railway gauge, spacing of parallel walls, fence posts, and structural beams.

How to Use This Calculator

  1. Enter A, B (same for both lines) and C₁, C₂ for the two parallel lines Ax+By+C=0.
  2. Click a sample example to auto-fill and calculate.
  3. Click Calculate to get the perpendicular distance.
  4. Review the visualization and step-by-step solution.
  5. Use Copy Results to share.

Disclaimer: This calculator is for parallel lines only. Both lines must have the same A and B coefficients.

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