Perpendicular Lines

There are two related situations:

• the given point lies away from the line, so the perpendicular must be dropped from the point to the line;
• the given point lies on the line, so the perpendicular must pass through that point.

In both constructions, pairs of equal compass arcs create points that are equally far from two points on the original line. Joining the appropriate points then produces a perpendicular bisector.

Constructing from a point away from a line

Suppose $P$ is the given point and does not lie on the given line.

1. Put the compass point at $P$. Choose a radius large enough for an arc to cut the given line in two places. Label these intersections $A$ and $B$.
2. Without using measurements, place the compass point at $A$ and draw an arc on the opposite side of the line from $P$.
3. Keeping the same compass width, repeat from $B$ so that the two arcs cross. Label their intersection $Q$.
4. Use a ruler as a straightedge to draw the line through $P$ and $Q$. Mark the right angle where it meets the given line.

This works because $PA=PB$: the first arc was drawn with centre $P$. Also, $QA=QB$ because the arcs drawn from $A$ and $B$ used the same radius. Points $P$ and $Q$ therefore both lie on the perpendicular bisector of $AB$, so the line $PQ$ is perpendicular to the original line.

Constructing at a point on a line

Now suppose the given point $P$ lies on the line.

1. Put the compass point at $P$ and draw an arc that cuts the line on both sides of $P$. Label the intersections $A$ and $B$.
2. Open the compasses to a radius greater than $AP$.
3. With centre $A$, draw an arc above the line.
4. Without changing this compass width, draw another arc from $B$ so that the arcs intersect at $Q$.
5. Draw the straight line through $P$ and $Q$ and mark the right angle at $P$.

The first arc makes $AP=BP$, so $P$ is the midpoint of $AB$. The matching arcs make $QA=QB$. Therefore $PQ$ is the perpendicular bisector of $AB$ and meets the original line at $90^\circ$.

Why perpendicular distance is the shortest distance

The distance from a point to a line means the length of the shortest segment joining the point to any point on that line. That shortest segment is always perpendicular to the line.

Let the perpendicular from $P$ meet the line at $H$. Choose any different point $X$ on the line. Triangle $PHX$ is right-angled at $H$: $PH$ is one shorter side, while the sloping segment $PX$ is the hypotenuse. The hypotenuse is the longest side of a right-angled triangle, so $PX>PH$. Therefore every non-perpendicular segment from $P$ to the line is longer than $PH$.

This is why a question asking for the distance from a point to a line requires the perpendicular length, not an arbitrary sloping measurement.