Parallel and Perpendicular Lines Calculator

The Two Rules You Need

Finding parallel and perpendicular lines comes down to two slope rules, and then point-slope form does the rest.

• Parallel lines share a slope. If the original line has slope m, a parallel line also has slope m.
• Perpendicular lines have negative-reciprocal slopes. Flip the slope over and change its sign. The negative reciprocal of 2 is −1/2, and the negative reciprocal of −3/4 is 4/3.

Notice the neat check: multiply a slope by its perpendicular slope and you always get −1. For 2 and −1/2, that is 2 × (−1/2) = −1.

How to Find a Parallel Line

Keep the same slope, then pin the line to your point with point-slope form. For a line with slope 2 through the point (1, 4), the parallel line is y − 4 = 2(x − 1), which becomes y = 2x + 2.

How to Find a Perpendicular Line

Take the negative reciprocal for the slope, then use the same point. The perpendicular slope to 2 is −1/2, so through (1, 4) the perpendicular line is y − 4 = −1/2(x − 1), which becomes y = −1/2 x + 9/2.

Special Case: Horizontal and Vertical Lines

A horizontal line has slope 0. Its parallel is another horizontal line, but its perpendicular is vertical, written as x = a constant, because the negative reciprocal of 0 is undefined. The calculator handles this for you and shows the vertical equation.

Check the Algebra

Every answer above is written in point-slope form first. If you want to review that form on its own, read the point-slope formula guide, see how to find it, or convert the results with the main calculator.