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Parallel and perpendicular lines: compare their slopes

Use equal slopes and negative reciprocals, including the exceptions.

Slopes can tell you how two straight lines are related. Distinct nonvertical parallel lines have the same slope. Nonvertical perpendicular lines have slopes whose product is -1, provided both slopes are nonzero.

Parallel lines have matching slopes

The equations y = 2x + 1 and y = 2x − 4 both have slope 2 but different intercepts. They rise at the same rate and never meet. If both slope and intercept are equal, the equations describe the same line rather than two distinct parallel lines.

Perpendicular slopes are negative reciprocals

If one line has slope 2/3, a perpendicular line has slope -3/2. Flip the fraction and change its sign. Their product is (2/3) × (-3/2) = -1.

For a line of slope -4, the perpendicular slope is 1/4. The sign changes again: a falling line can be perpendicular to a rising line.

Build a perpendicular line through a point

Suppose the original slope is 2/3 and the new line must pass through (2, 1). The new slope is -3/2. Use point-slope form: y − 1 = (-3/2)(x − 2). Expanding gives y = (-3/2)x + 4.

Check the point: (-3/2) × 2 + 4 = 1. Check the direction relationship separately by multiplying the two slopes and confirming -1.

Horizontal and vertical exceptions

A horizontal line has slope 0. Its perpendicular is vertical, with undefined slope, so the negative-reciprocal formula cannot produce a finite number. Distinct vertical lines are parallel to each other; a vertical and a horizontal line meet at a right angle.

A note about the graph

These slope rules use ordinary Cartesian coordinates with equal geometric units on the axes. A screenshot with stretched axes can make a right angle appear distorted. Use the equations to check the relationship rather than judging the display alone.

Use the slope calculator separately for each pair of points, then compare the exact fractions. Review point-slope form to construct the new line and special slopes for the horizontal and vertical cases.