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Parallel & Perpendicular Line Calculator with Graph

Geometry & Coordinate GeometryFreeStep by stepGraph

Parallel lines have the same slope, and perpendicular lines have slopes that multiply to −1, so the perpendicular slope is the negative reciprocal. This calculator takes a line y = mx + c and a point, then finds the parallel and perpendicular lines through that point and graphs all three, including the vertical line case.

Try:

How to use the parallel & perpendicular line calculator

  1. Enter the slope m and intercept c of the given line.
  2. Enter the point the new lines must pass through.
  3. Press Calculate to see both equations and the graph.

Formula

m∥=m,m⊥=−1mm_{\parallel} = m,\quad m_{\perp} = -\frac{1}{m}

Worked example: y = 2x + 1 through (4, −1)

  • Slope of the given line (m): 2
  • Its y-intercept (c): 1
  • Point x₁: 4
  • Point y₁: -1

✓ Answer checked

Parallel: y=2x−9,Perpendicular: y=−12x+1\text{Parallel: } y = 2x - 9,\quad \text{Perpendicular: } y = -\frac{1}{2}x + 1
Parallel slope
22
Perpendicular slope
−12-\frac{1}{2}
−20−15−10−5510152025−15−10−551015(4, -1)given lineparallelperpendicular
Parallel lines have equal slopes; perpendicular lines meet at a right angle (slopes multiply to −1).
Step-by-step working (3 steps)

  1. Parallel slope

    Parallel lines have the same slope.

    m∥=2m_{\parallel} = 2

  2. Perpendicular slope m₁ × m₂ = −1

    Take the negative reciprocal: flip the fraction and change the sign.

    m⊥=−12=−12m_{\perp} = -\frac{1}{2} = -\frac{1}{2}

  3. Use the point

    Find each intercept with c = y₁ − m·x₁.

    y=2x−9,y=−12x+1y = 2x - 9,\quad y = -\frac{1}{2}x + 1
Formulas used
m⊥=−1mm_{\perp} = -\frac{1}{m}
How this was checked
  • ✓ Both new lines pass through the point, and the slopes multiply to −1.

Frequently asked questions

What is the perpendicular slope to 2?

−1/2. Flip the fraction and change the sign.

How do I find a parallel line through a point?

Keep the slope and find c from c = y₁ − m x₁.

What is perpendicular to a horizontal line?

A vertical line, x = constant, whose slope is undefined.

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