Perpendicular bisector of a segment

The perpendicular bisector of a line segment $\overline{AB}$ is the line that meets $\overline{AB}$ at its midpoint and is perpendicular to it. Drawing one is one of the most fundamental ruler-and-compass constructions, and it leads to many others.

Concept

What we want to build

Given a segment $\overline{AB}$, we want a line $\ell$ such that:

1. $\ell$ passes through the midpoint $M$ of $AB$.
2. $\ell$ is perpendicular to $\overline{AB}$ at $M$.

The construction

1. Draw the segment $\overline{AB}$.
2. With the compass tip at $A$ and a radius larger than half of $AB$, draw an arc above $\overline{AB}$ and another below.
3. Without changing the compass spread, place the tip at $B$ and draw arcs above and below.
4. The two pairs of arcs intersect at two points , call them $P$ (above) and $Q$ (below).
5. Draw the line $\overleftrightarrow{PQ}$ using the ruler. That is the perpendicular bisector.

Why does it work?

Every point on $\overleftrightarrow{PQ}$ is the same distance from $A$ as from $B$. The arcs we drew were all of the same radius, so $PA = PB$ and $QA = QB$. A point that is equidistant from the endpoints of a segment must lie on the perpendicular bisector , and conversely, every point on the perpendicular bisector is equidistant from the endpoints. This is the defining property of the perpendicular bisector.

So a beautiful way to think about it: the perpendicular bisector of $\overline{AB}$ is the set of all points equidistant from $A$ and $B$.

Why "larger than half"?

If your compass spread is less than half of $AB$, the arcs from $A$ and $B$ will not meet. They have to overlap to give intersection points. Any spread larger than half $AB$ works , and equal spreads at both ends are essential.

Useful corollaries

• Once you have the perpendicular bisector, you also have the midpoint of $AB$ (the point where the bisector meets $AB$).
• You can find the centre of a chord of a circle by drawing its perpendicular bisector.
• The perpendicular bisectors of all three sides of a triangle meet at a single point called the circumcentre , the centre of the circle passing through all three vertices.

Worked examples

Example 1. Construct the perpendicular bisector of a segment $AB$ of length $6$ cm.

• Draw $\overline{AB}$, $6$ cm.
• Open the compass to about $4$ cm (more than $3$ cm).
• Arc above and below from $A$, then from $B$.
• Connect the two intersection points $P$ and $Q$ to get the perpendicular bisector.
• The midpoint is where $PQ$ crosses $AB$.

Example 2. Find the midpoint of a segment $\overline{XY}$ of length $7$ cm.

• Construct its perpendicular bisector. Where it crosses $\overline{XY}$ is the midpoint.

Example 3. Show that any point on the perpendicular bisector of $\overline{AB}$ is equidistant from $A$ and $B$.

• Let $P$ be any point on $\overleftrightarrow{PQ}$.
• By construction, the arcs from $A$ and $B$ had the same radius, so $PA = PB$.

Example 4. Given a circle and a chord, find the centre using a perpendicular bisector.

• Draw the chord.
• Construct its perpendicular bisector.
• The bisector passes through the centre of the circle. (To find the centre exactly, draw a second chord and its perpendicular bisector , the two bisectors meet at the centre.)

Try it yourself

1. Draw a segment of $8$ cm and construct its perpendicular bisector. Measure to confirm the midpoint.
2. Construct the perpendicular bisector of a vertical segment.
3. Construct the perpendicular bisector of a slanted segment.
4. Draw a triangle. Construct the perpendicular bisector of each side. Do all three meet at one point?
5. Construct an isosceles triangle by starting with a base and using the perpendicular bisector to place the third vertex.
6. Use a perpendicular bisector to divide a segment into two equal parts, then divide one of those parts into two more equal parts. The original segment is now in what ratio?
7. Given a circle, find its centre using two chords and their perpendicular bisectors.
8. Investigate: what is the perpendicular bisector of a diameter of a circle? Where does it pass through?

Activity

The crossroads point. Mark two points $A$ and $B$ on a sheet. Imagine a treasure is buried somewhere equidistant from $A$ and $B$. Construct the perpendicular bisector of $\overline{AB}$. Mark $3$ random points along the bisector. With a ruler, measure each point's distance to $A$ and to $B$. The two distances should always be equal , your perpendicular bisector is correct.