Bisectors

The perpendicular bisector of $AB$ is the whole line that passes through $M$ at a right angle to $AB$. Its name describes both of its properties:

• bisector: it divides $AB$ into two equal lengths;
• perpendicular: it meets $AB$ at $90°$.

A line through the midpoint but at another angle is only a bisector. A perpendicular line that crosses somewhere other than the midpoint is not a bisector.

Constructing it with a ruler and compasses

Begin with the given line segment $AB$. Use the ruler as a straightedge to draw lines, not to locate the midpoint by measurement.

1. Open the compasses to a width greater than half the length of $AB$.
2. Place the compass point at $A$. Draw an arc above $AB$ and another below it.
3. Without changing the compass width, place the compass point at $B$. Draw arcs that cross the first pair. Label the two intersections $P$ and $Q$.
4. Use a ruler to draw the straight line $PQ$.
5. Label the point where $PQ$ crosses $AB$ as $M$.

The line $PQ$ is the perpendicular bisector, and $M$ is the constructed midpoint. Keep the arcs on the diagram because they are part of the construction.

Why the construction works

Every arc was drawn with the same compass width. Consequently,

$AP=BP$

and

$AQ=BQ.$

So both $P$ and $Q$ are equally distant from $A$ and $B$. Joining these two points produces the line of points equidistant from $A$ and $B$: their perpendicular bisector.

Where this line crosses $AB$, at $M$, it divides the segment equally, so $AM=MB$. It also meets $AB$ at $90°$. This explains why the construction finds the midpoint and creates a perpendicular at the same time.

Why the compass width matters

The radius must be greater than half of $AB$. If it is exactly half, the arcs merely meet at the midpoint, giving only one intersection. If it is smaller, they do not meet at all. A larger radius produces two clear intersections, one on each side of the segment, so the bisector can be drawn accurately.

The exact radius does not matter: what matters is that it is sufficiently large and remains unchanged while all four arcs are drawn.