How to Find the Distance From a Point to a Line

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A good grasp of algebra will help you solve geometry problems such as finding the distance from a point to a line. The solution involves creating a new perpendicular line joining the point to the original line, then finding the point where the two lines intersect, and finally calculating the length of the new line to the point of intersection.

TL;DR (Too Long; Didn't Read)

To find the distance from a point to a line, first find the perpendicular line passing through the point. Then using the Pythagorean theorem, find the distance from the original point to the point of intersection between the two lines.

Find the Perpendicular Line

The new line will be perpendicular to the original one, that is, the two lines intersect at right angles. To determine the equation for the new line, you take the negative inverse of the slope of the original line. Two lines, one with a slope A, and the other with a slope, -1÷A, will intersect at right angles. The next step is to substitute the point into the equation of slope-intercept form of new line to determine its y-intercept.

As an example, take the line y = x + 10 and the point (1,1). Note that the slope of the line is 1. The negative reciprocal of 1 is -1÷1 or -1. So the slope of the new line is -1, so the slope-intercept form of the new line is y = -x + B, where B is a number you don’t yet know. To find B, substitute the x and y values of the point into the line equation:
y = -x + B

Use the original point (1,1), so substitute 1 for x and 1 for y:

1 = -1 + B1 + 1 = 1 - 1 + B add 1 to both sides2 = B

You now have the value for B.

The new line’s equation then is y = -x + 2.

Determine Intersection Point

The two lines intersect when their y values are equal. You find this by setting the equations equal to each other, then solve for x. When you’ve found the value for x, plug the value into either line equation (it doesn’t matter which one) to find the point of intersection.

Continuing the example, you have the original line:
y = x + 10
and the new line, y = -x + 2
x + 10 = -x + 2 Set the two equations equal to each other.
x + x + 10 = x -x + 2 Add x to both sides.
2x + 10 = 2
2x + 10 - 10 = 2 - 10 Subtract 10 from both sides.
2x = -8
(2÷2)x = -8÷2 Divide both sides by 2.
x = -4 This is the x value of the intersection point.
y = -4 + 10 Substitute this value for x into one of the equations.
y = 6 This is the y value of the intersection point.
The intersection point is (-4, 6)

Find Length of a New Line

The length of the new line, between the given point and the newly-found intersection point, is the distance between the point and the original line. To find the distance, subtract the x and y values to get the x and y displacements. This gives you the opposite and adjacent sides of a right triangle; the distance is the hypotenuse, which you find with Pythagorean theorem. Add the squares of the two numbers, and take the square root of the result.

Following the example, you have the original point (1,1) and the point of intersection (-4,6).
x1 = 1, y1 = 1, x2 = -4, y2 = 6
1 - (-4) = 5 Subtract x2 from x1.
1 - 6 = -5 Subtract y2 from y1.
5^2 + (-5)^2 = 50 Square the two numbers, then add.
√ 50 or 5 √ 2 Take the square root of the result.
5 √ 2 is the distance between the point (1,1) and the line, y = x + 10.

References

About the Author

Chicago native John Papiewski has a physics degree and has been writing since 1991. He has contributed to "Foresight Update," a nanotechnology newsletter from the Foresight Institute. He also contributed to the book, "Nanotechnology: Molecular Speculations on Global Abundance." Please, no workplace calls/emails!