Distance Between Two Parallel Lines
Overview
The distance from a point to a line is not measured along just any segment — it is the length of the perpendicular segment from the point to the line. Finding it is a four-step routine: get the slope of the given line, write the equation of the perpendicular line through the point, solve the two equations as a system to find the foot of the perpendicular, and finish with the distance formula. The same routine, with one extra step at the start, gives the distance between two parallel lines.
Vocabulary
- Distance from a Point to a Line
- The length of the perpendicular segment drawn from the point to the line. It is the shortest distance between the point and the line.
- Foot of the Perpendicular
- The point where the perpendicular segment from a given point intersects the given line; it is found by solving the system formed by the two lines' equations.
- Distance Between Two Parallel Lines
- The length of a perpendicular segment joining the two lines. It is the same everywhere, and it is found by measuring the distance from any point on one line to the other line.
- Distance Formula
- The distance between two points (x1, y1) and (x2, y2): the square root of ((x2 - x1) squared + (y2 - y1) squared).
- System of Equations
- Two or more equations considered together; the solution is the point that satisfies all of them — here, the intersection point of the given line and the perpendicular line.
Main Ideas
- The four steps for point-to-line distance: (1) determine the slope of the given line; (2) write the equation of the line perpendicular to it through the given point, using the opposite reciprocal slope; (3) solve the system of the two equations to find where they intersect; (4) use the distance formula between the given point and that intersection point.
- The perpendicular direction is essential: distance to a line is always measured along a perpendicular segment, never along a slanted one.
- For the distance between two parallel lines, first pick any point on the first line (setting x = 0 and solving for y is usually easiest), then run the same four steps from that point to the second line.
- Expect answers that are not whole numbers: distances in this lesson are typically reported as decimals rounded to the nearest tenth.
Lesson Notes
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Interactive Activity
Find the distance from a point to a line and between two parallel lines.
Distance Between Two Parallel Lines — Interactive ActivityOpen ›
Practice Problems
Want More Practice?
Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.