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Parallel and Perpendicular Line Calculator

Enter a line by two points and a third point, and get the parallel and perpendicular lines through it.

What do you need?
Try:

The two points define the line. The new lines pass through the extra point.

Parallel line through the point

y=2x−1y=2x-1y=2x−1
Slope m
222
Angle with the x axis
63.4349°
Slope-intercept form
y=2x+1y=2x+1y=2x+1
Point-slope form
y−1=2(x)y-1=2(x)y−1=2(x)
Standard form
2x−y=−12x-y=-12x−y=−1
Rise and run
8 and 4
Distance
8.944272
Midpoint
(2, 5)

Through the extra point

Parallel line
y=2x−1y=2x-1y=2x−1
Perpendicular line
y=−12x+4y=-\tfrac{1}{2}x+4y=−21​x+4

Graph

  • Line
  • Parallel
  • Perpendicular
Table of values
Graph of the line through (0, 1) and (4, 9): sampled values across the visible range
xLineParallelPerpendicular
-5.6-10.2-12.26.8
-4.08-7.16-9.166.04
-2.56-4.12-6.125.28
-1.04-1.08-3.084.52
0.481.96-0.043.76
2533
3.528.046.042.24
5.0411.089.081.48
6.5614.1212.120.72
8.0817.1615.16-0.04
9.620.218.2-0.8

Step by step

  1. 1Slope
    m=y2−y1x2−x1m=\dfrac{y_2-y_1}{x_2-x_1}m=x2​−x1​y2​−y1​​
    ↓
    m=9−14−0=84=2m=\dfrac{9-1}{4-0}=\dfrac{8}{4}=2m=4−09−1​=48​=2
  2. 2Parallel line
    m∥=mm_\parallel=mm∥​=m
    ↓
    y−3=2(x−2)  ⇒  y=2x−1y-3=2(x-2)\;\Rightarrow\;y=2x-1y−3=2(x−2)⇒y=2x−1

    A parallel line has the same slope.

  3. 3Perpendicular line
    m⊥=−1mm_\perp=-\dfrac{1}{m}m⊥​=−m1​
    ↓
    m⊥=−12=−12  ⇒  y=−12x+4m_\perp=-\dfrac{1}{2}=-\tfrac{1}{2}\;\Rightarrow\;y=-\tfrac{1}{2}x+4m⊥​=−21​=−21​⇒y=−21​x+4

    A perpendicular line has the negative reciprocal slope.

  1. Tools
  2. Mathematics
  3. Slope Calculator
  4. Parallel and Perpendicular Line Calculator

About Parallel and Perpendicular Line Calculator

This calculator finds the line through a chosen point that is parallel to a given line, and the one that is perpendicular to it. The given line is defined by two points, and all three lines are drawn on one graph. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

A parallel line keeps the same slope. A perpendicular line uses the negative reciprocal. For the line through (0, 1) and (4, 9), slope 2, and the point (2, 3), the parallel is y=2x-1.

How to use Parallel and Perpendicular Line Calculator

  1. 1Type the coordinates of point 1 and point 2. Negative numbers and decimals are fine.
  2. 2Pick what you need: everything, slope and angle, line equation, distance, midpoint, or parallel and perpendicular lines.
  3. 3Read the headline answer and the table of every result under it.
  4. 4Check the graph. Drag to pan and scroll or pinch to zoom, and use the trace to read coordinates.
  5. 5Open the step-by-step working to copy the method into your homework.
  6. 6For parallel and perpendicular lines, enter the point the new lines must pass through.

Features

  • Slope as an exact fraction and a decimal, with rise and run.
  • Angle of inclination between 0 and 180 degrees.
  • Slope-intercept, point-slope and standard form with whole-number coefficients.
  • Vertical and horizontal lines handled correctly (slope undefined or zero).
  • Distance formula and midpoint formula with steps.
  • Parallel and perpendicular lines through a chosen point.
  • An interactive graph of the line, the points and the midpoint.
  • Step-by-step working in KaTeX.

Slopes of related lines

Parallel lines share a slope, and perpendicular slopes multiply to −1.

m∥ = m
m⊥ = −1 / m

Worked examples

  • Line (0, 1), (4, 9); point (2, 3) → gives Perpendicular slope -0.5

    The negative reciprocal of 2 is −1/2.

Privacy: how it works

  • Every result is calculated by code in your browser tab. The coordinates you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.

Tips

  • Keep the order consistent. Subtract the first point from the second in both the numerator and the denominator, or the sign of the slope flips.
  • If both points are the same, no line is defined. The calculator tells you.
  • Two lines are perpendicular when the product of their slopes is −1, and parallel when the slopes are equal.
  • Standard form is scaled to whole numbers with no common factor, so decimals in the points are cleared for you.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Your points are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Your points are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

How do I find a parallel line through a point?

Use the same slope as the given line and write the point-slope form with your point.

What if the given line is horizontal?

The parallel is also horizontal, and the perpendicular is the vertical line x = the point’s x.

More Slope Calculator pages

  • Slope CalculatorEnter two points to get the slope, the angle, the equation of the line, the distance and the midpoint, with the working and a graph.
  • Slope From Two Points CalculatorEnter two points and get rise over run and the angle of the line.
  • Distance Formula CalculatorEnter two points to get the straight-line distance between them.
  • Midpoint CalculatorEnter two endpoints to get the midpoint of the segment.
  • Line Equation From Two PointsEnter two points to get the equation of the line in three forms.
  • Slope-Intercept Form CalculatorEnter two points and read the slope and the y-intercept of y = mx + b.
  • Perpendicular Line CalculatorEnter a line by two points and the point the perpendicular must pass through.

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