1. Perpendicular Bisector of a Segment
A perpendicular bisector is a line that divides a line segment into two equal halves at an exact right angle ($90^\circ$).
Steps of Construction:
- Given segment $AB$, open the compass to a radius strictly greater than half of $AB$.
- With center $A$, draw two arcs—one above $AB$ and one below $AB$.
- With center $B$ and the exact same radius, draw two arcs intersecting the previous arcs at points $P$ and $Q$.
- Draw the straight line joining $P$ and $Q$. Line $PQ$ is the perpendicular bisector of $AB$!
2. Bisector of an Angle
An angle bisector is a ray that cuts an angle into two equal halves ($\angle 1 = \angle 2 = \frac{1}{2}\angle \text{Original}$).
Steps of Construction:
- Given $\angle ABC$, with center $B$, draw an arc cutting arm $BA$ at $P$ and arm $BC$ at $Q$.
- With center $P$ and radius $> \frac{1}{2}PQ$, draw an arc inside the angle.
- With center $Q$ and the same radius, draw an arc intersecting the previous arc at $R$.
- Draw ray $BR$. Ray $BR$ bisects $\angle ABC$!
4. Pitfall & Examiner Trap
⚠️ Trap: Opening Compass Less Than Half the Segment
If the compass radius is less than or equal to half of $AB$, the arcs drawn from $A$ and $B$ will never meet! Always ensure radius $> \frac{1}{2}AB$.
If the compass radius is less than or equal to half of $AB$, the arcs drawn from $A$ and $B$ will never meet! Always ensure radius $> \frac{1}{2}AB$.
5. Why This Matters in Life
Road planners use perpendicular bisectors to find the ideal location for a hospital equidistant from two neighboring towns.