How to Find the Equation of a Perpendicular Line Given an Equation and Point

Co-authored by Grace Imson, MA

Last Updated: December 10, 2020

Equations of perpendicular lines are usually introduced in the beginning of geometry or algebra, and are the starting points of many mathematical concepts. Some students may find them complex, but with this guide, you can find perpendicular lines with ease!

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In this guide, the slope would be m in slope-intercept form (y=mx+b).^{[1]
X
Expert Source

Grace Imson, MA
Math Instructor, City College of San Francisco

Expert Interview} The photo above identifies 2/3 as the slope. The slope does not have to be fraction; it can be a whole number.
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Change the slope. To change the slope, you must convert the value into its opposite sign (positive to negative or negative to positive). Plus, it must be put into its reciprocal version. The order in which the conversion is done does not matter. Refer to the example above.
- 2/3 becomes -2/3. This makes the slope opposite.
- -2/3 becomes -3/2. This makes the slope both opposite and reciprocal. Thus, the slope has been converted.
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Replace the old slope with the new slope. Replace the y-intercept's value with a variable (b).
This will make the equation ready to be solved. Solving it will lead to the y-intercept's value being found.^{[2]
X
Expert Source

Grace Imson, MA
Math Instructor, City College of San Francisco

Expert Interview}
Multiply the new slope with the x-value. Then, cancel out the product (make it become 0) with either addition or subtraction. Don't forget to add or subtract the y-value too. In the end, you should get the y-intercept.
Still using slope-intercept form, use the new slope and the new y-intercept's value. This is the final answer.
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(Optional) Check whether or not your answer is correct. Graph the two equations and measure one of the angles that forms; according to the definition of a perpendicular line, all four angles have to measure 90 degrees.
- If the lines are horizontal and vertical, then they are perpendicular due to the "squares" of the coordinate grid.

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How do I find the perpendicular bisector between two points?

Donagan

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See Find the Perpendicular Bisector of Two Points.

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How do you find a perpendicular slope when the slope given is negative. For example, if asked: What is the slope of a line perpendicular to -4/5?" What would my answer be?

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You use the negative reciprocal, in this case +5/4.

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