Distance from a Point to a Line

There are infinitely many ways to travel from a point to a line, but only one of them is shortest. Picture a straight shoreline and an island offshore: to build the cheapest bridge you would not lay it at an angle — you would run it perpendicular to the shore.

Concept The shortest path is perpendicular

The distance from a point to a line is the length of the perpendicular segment from the point to that line.

Any slanting path is longer. Only the perpendicular gives the true distance, so "distance" always means the perpendicular length.

Concept Distance between two parallel lines

The same idea extends to a pair of parallel lines: the distance between them is the perpendicular distance, measured from any point on one line across to the other.

Because the lines are parallel, that measurement is constant. It does not matter whether you measure near the start, the middle or the far end — the answer is the same every time.

Concept A theorem that runs the other way

If two lines in the same plane are everywhere the same distance from a third line, then those two lines are parallel.

The reasoning is straightforward. Holding the same distance from a reference line at every point means neither line is drifting towards or away from it — so their directions must match, and equal directions mean equal slopes.

If instead the gap widens or narrows as you move along, the slopes differ and the lines are not parallel.

Example The bridge problem

A straight shoreline runs east to west, and an island sits offshore. Where should the bridge be built to keep it as short as possible?

A bridge laid at an angle covers extra sideways distance for no gain.

Shortening it means removing that sideways component entirely.

What remains is the perpendicular from the island to the shore.

⟹ build the bridge perpendicular to the shoreline

Example Checking whether lines are parallel

Two lines are measured against a third. The first stays 3 units away all along. The second is 3 units away at one end but 5 units away at the other. What can you conclude?

The first keeps a constant distance, so it is parallel to the reference line.

The second's distance changes, so its direction differs.

⟹ only the first line is parallel to the reference line

A changing gap is a reliable signal that two lines will eventually meet.

Note Mistakes to avoid

Measuring a slanting segment and calling it the distance.

Forgetting that the perpendicular must pass through the given point.

Expecting the gap between non-parallel lines to be a single fixed number.

Measuring between two parallel lines at an angle rather than perpendicular.

Overlooking that the two lines must lie in the same plane for the theorem to apply.

Summary

The distance from a point to a line is the length of the perpendicular from the point to the line.
Any other path is longer, which is why the perpendicular defines the distance.
Between two parallel lines the perpendicular distance is constant everywhere.
Two lines that stay the same distance from a third line are parallel to each other.
A changing gap means different slopes, so the lines are not parallel.