The distance from a point to a line is the shortest straight-line separation between them. In geometry, the shortest path from a point to a line always meets the line at a right angle. This idea matters in coordinate geometry, physics, engineering, computer graphics, and optimization because it turns a visual measurement into a formula.
It lets you find the exact distance without drawing a scale diagram.
Key Facts
- For a line Ax + By + C = 0 and point P(x1, y1), the distance is d = |Ax1 + By1 + C| / sqrt(A^2 + B^2).
- The shortest segment from a point to a line is perpendicular to the line.
- The numerator |Ax1 + By1 + C| measures how far the point is from satisfying the line equation.
- The denominator sqrt(A^2 + B^2) normalizes the result using the length of the line's normal vector.
- If Ax1 + By1 + C = 0, then the point lies on the line and d = 0.
- For a horizontal line y = k, the distance from (x1, y1) is |y1 - k|; for a vertical line x = h, the distance is |x1 - h|.
Vocabulary
- Perpendicular distance
- The perpendicular distance is the shortest distance from a point to a line, measured along a segment that forms a 90 degree angle with the line.
- Standard form
- Standard form of a line is Ax + By + C = 0, where A, B, and C are constants and A and B are not both zero.
- Normal vector
- A normal vector is a vector perpendicular to a line, and for Ax + By + C = 0 it can be written as (A, B).
- Foot of the perpendicular
- The foot of the perpendicular is the point where the shortest segment from the point meets the line.
- Absolute value
- Absolute value gives the nonnegative size of a number, which is why distance is never negative.
Common Mistakes to Avoid
- Using the slope distance instead of the perpendicular distance. The shortest distance must meet the line at a 90 degree angle, not follow a slanted or horizontal path unless the line makes that correct direction.
- Forgetting to put the line in Ax + By + C = 0 form. The formula d = |Ax1 + By1 + C| / sqrt(A^2 + B^2) only works when all terms are on one side and the other side is 0.
- Leaving out the absolute value in the numerator. A signed result can be negative, but distance must always be nonnegative.
- Dividing by A^2 + B^2 instead of sqrt(A^2 + B^2). The denominator is the length of the normal vector, so it must use the square root.
Practice Questions
- 1 Find the distance from P(3, 4) to the line 2x - y + 5 = 0.
- 2 Find the distance from P(-2, 7) to the line 3x + 4y - 12 = 0.
- 3 A student says the distance from a point to a line can be found by drawing any segment from the point to the line and measuring it. Explain why this is incorrect and identify which segment gives the true distance.