Bisectors

Step-by-step Construction

How do you build this perfect dividing line? Follow this sequence precisely to construct a perpendicular bisector:

• First, place the sharp point (needle) of your compass on the first endpoint of the line segment (let's call it $A$). Open the compass so it is visibly past the halfway point.
• Second, draw a large, clear arc that extends both well above and well below the line segment.
• Third, without changing the compass width at all, lift the compass and place the needle on the second endpoint ($B$).
• Fourth, draw another large arc. It must cross your first arc at two distinct intersection points—one above the line segment and one below it.
• Finally, use your straight edge and a sharp pencil to draw a solid straight line connecting those two intersection points.

Identifying the Midpoint

The perpendicular bisector does more than just split the area; it pinpoints a specific location. The exact spot where your constructed bisector crosses the original line segment is the precise midpoint.

To verify your construction, use a ruler to measure the distance from endpoint $A$ to the midpoint, and from endpoint $B$ to the midpoint. If your construction is accurate, these two distances will be identical.

Equidistant Properties and Loci

Imagine trying to map out all the possible locations that are exactly halfway between two rival towns. In geometry, a set of points that follows a specific rule like this is called a locus (plural loci).

The perpendicular bisector is actually the locus of all points that are equidistant from two fixed points. Every single dot along the perpendicular bisector is exactly the same distance from endpoint $A$ as it is from endpoint $B$.

If an exam question asks you to "shade the region closer to point $A$ than point $B$", you must construct the perpendicular bisector between them. Then, shade the entire side of the paper where point $A$ is located.

Worked Example

Construct the perpendicular bisector of a line segment $PQ$ that is $9\text{ cm}$ long, and label its midpoint $M$.

Step 1: Draw the initial line segment.

• Draw a straight line $PQ$ measuring exactly $9\text{ cm}$.

Step 2: Draw the first arc.

• Place the compass needle on point $P$. Open the compass to roughly $6.5\text{ cm}$ (this must be greater than half of $9\text{ cm}$, which is $4.5\text{ cm}$). Draw a large arc above and below the line.

Step 3: Draw the second arc.

• Keeping the compass open to exactly $6.5\text{ cm}$, move the needle to point $Q$. Draw another arc so that it crosses the first arc at two points.

Step 4: Draw the bisector.

• Use a straight edge to draw a line directly through the two points where the arcs intersect.

Step 5: Mark and verify the midpoint.

• Mark the exact point where the bisector crosses line $PQ$ with the letter $M$. Measure the distance from $P$ to $M$; it should be exactly $4.5\text{ cm}$.