Perpendicular and Parallel Line Calculator

Calculator

Choose an operation and enter line data

All fields accept decimals and negatives.

Operation

Line 1 format

Line 2 format

Line 1 inputs

Enter only the fields for the selected format.

A

B

C

m

b

x1

y1

x2

y2

Line 2 inputs

Only required for the selected operation.

A

B

C

m

b

x1

y1

x2

y2

Point inputs

Required for “through a point” and “point-to-line distance”.

x0

y0

Reset Results appear above the form after calculation.

Example data

Sample inputs and expected outputs

Scenario	Inputs	Key outputs
Through a point	Line: y = 2x + 1, Point: (1, 2)	Parallel: y = 2x + 0, Perpendicular: y = -0.5x + 2.5
Relationship	Line1: x − y = 0, Line2: x − y − 2 = 0	Parallel, no intersection, distance = 1.4142…
Point distance	Line: 3x + 4y − 10 = 0, Point: (2, 1)	Distance = 0.4, Foot ≈ (2.24, 1.32)

These are reference examples; your results will update from your inputs.

Formulas used

Core equations behind the results

Standard form: Ax + By + C = 0.
Slope from standard: m = −A/B (when B ≠ 0).
Parallel slopes: m_parallel = m.
Perpendicular slopes: m_perp = −1/m (special cases: vertical ↔ horizontal).
Line through a point: y − y0 = m(x − x0), or vertical x = x0.
Intersection (Cramer’s rule): solve two linear equations for (x, y).
Point-to-line distance: d = |Ax0 + By0 + C| / √(A² + B²).
Foot of perpendicular: (x, y) = (x0, y0) − (A, B) * (Ax0 + By0 + C)/(A² + B²).
Angle between lines: uses direction vectors (B, −A) and cosθ = |d1·d2|/(|d1||d2|).

How to use

Steps to get accurate results

Select an operation at the top.
Choose the format for each line you will enter.
Fill only the fields for the chosen format (others can be ignored).
When required, provide the point (x0, y0).
Press Calculate to view results above the form.
Use Download CSV or Download PDF to save outputs.

Article

1) What this line calculator solves

This tool works with straight lines in a coordinate plane. It focuses on parallel and perpendicular relationships, and it can also compute intersections, acute angles, point‑to‑line distance, and distance between parallel lines. Use it to verify analytic geometry steps or check answers fast. It is useful for drafting and tutoring too.

2) Input formats supported

Enter a line in standard form (Ax + By + C = 0), slope‑intercept form (y = mx + b), or by two points ((x1, y1) and (x2, y2)). Each input is converted to standard form so vertical and horizontal lines are handled consistently.

3) Parallel line facts and data

Parallel lines never meet and keep a constant separation. For non‑vertical lines, they share the same slope m. For vertical lines, both are x = constant. The calculator can build a parallel line through a chosen point and show both equation styles for copying.

4) Perpendicular line facts and data

Perpendicular lines form a 90° angle. When slopes are finite, the rule is m1 · m2 = −1, so the perpendicular slope becomes −1/m. A vertical line is perpendicular to a horizontal line. The tool automatically applies these rules and generates the new equation through your point.

5) Intersection and angle results

If two lines are not parallel, they intersect at exactly one point. The calculator solves the 2×2 system to return (x, y). It also reports the acute angle using direction vectors, which stays reliable when one line is vertical or nearly vertical.

6) Distance computations

Point‑to‑line distance uses d = |Ax0 + By0 + C| / √(A² + B²). For parallel lines, the distance comes from normalized coefficients, giving the constant gap in the same units as x and y. The foot of the perpendicular is also listed.

7) Practical uses and accuracy tips

Keep units consistent and avoid rounding inputs too early. Let the calculator compute full precision, then round outputs for your final report. Test with the example table to understand edge cases like vertical lines, horizontal lines, and very small slopes.

FAQs

1) How do I know if two lines are parallel?

Two lines are parallel when their direction is the same. The calculator checks proportional coefficients in standard form, and also compares slopes when defined. If parallel, it will show “Parallel” and no single intersection point.

2) How can I enter a vertical line?

Use standard form with B = 0, like 1x + 0y − 5 = 0, which represents x = 5. You can also create a vertical line by choosing two points that share the same x value.

3) Why is the perpendicular slope −1/m?

For non‑vertical lines, perpendicular lines satisfy m1·m2 = −1, which rearranges to m2 = −1/m1. The tool applies special cases automatically: vertical is perpendicular to horizontal, and vice versa.

4) What does point‑to‑line distance represent?

It is the shortest distance from your point to the line, measured along a perpendicular segment. The calculator also reports the foot of that perpendicular, which is the exact point where the shortest segment meets the line.

5) What happens if my two lines are the same line?

If coefficients are proportional including the constant term, the calculator labels them “Coincident.” That means infinitely many intersection points, and the distance between the lines is zero.

6) How do the CSV and PDF downloads work?

After you calculate, the page stores the latest result and enables download buttons. CSV exports key–value rows for spreadsheets. PDF produces a clean one‑page summary you can attach to homework or reports.