How to Find a Line Perpendicular to Another

By Bryan Keythman
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Two lines are perpendicular to one another if they intersect each other in the shape of a "t" to form a 90-degree angle. A line that is perpendicular to another has a slope that is negative and reciprocal of the slope of the other line. A reciprocal is a number that is turned upside down, with its numerator switched with its denominator. For example, a line with a slope of 3/4 is perpendicular to a line with a slope of -4/3. You can determine the equation of a line perpendicular to another line if you know the equation of one line and the y-intercept of the other.

Determine the known equation of a line. For the following example, use the equation y - 2 = (1/2)x.

Arrange the equation into its slope-intercept form, which is y = mx + b. In the slope intercept form, "m" represents the slope and "b" represents the y-intercept, which is the point where the line crosses the y-axis. In the example, add 2 to both sides of the equation, which results in y - 2 + 2 = (1/2)x + 2. This leaves y = (1/2)x + 2, which is the line written in its slope-intercept form.

Determine the number that's in the "m" position of the slope-intercept form of the line's equation to determine its slope. In the example, 1/2 is in the "m" position, so 1/2 is the line's slope.

Move the numerator to the denominator position and the denominator to the numerator position of the slope and make the result the opposite sign of its current sign to determine its negative reciprocal. In the example, move 1 to the denominator and 2 to the numerator and place a negative sign in front of it, which equals -2/1. This leaves -2, which is the slope of the unknown line that's perpendicular to the first line.

Determine the known y-intercept of a line that's perpendicular to the first line. In the example, use a y-intercept of -1.

Substitute the unknown line's slope into the "m" position and its y-intercept into the "b" position of the slope intercept form of the line's equation to determine the equation of the line that's perpendicular to the first. In the example, substitute the slope of -2 into the "m" position and the y-intercept of -1 into the "b" position, which results in y = -2x - 1. This is the equation of a line that's perpendicular to the first.