Properties of the Perpendicular Bisector of a Segment
What is a Perpendicular Bisector?
A perpendicular bisector of a line segment is a line that:
- Passes through the midpoint of the segment (bisects it into two equal parts)
- Is perpendicular (forms a 90° angle) to the segment
If we have a segment $\overline{AB}$, its perpendicular bisector is a line $\ell$ such that:
$$\ell \perp \overline{AB} \quad \text{and} \quad AM = MB$$
where $M$ is the midpoint of $\overline{AB}$.
Key Vocabulary
- 📐 Perpendicular: Meeting at a 90° angle
- ✂️ Bisect: To cut into two equal parts
- 📍 Midpoint: The exact center point of a segment
- 📏 Equidistant: At equal distances
- 🔄 Locus: A set of all points satisfying a condition
⭐ The Fundamental Theorem (Most Important Property!)
Perpendicular Bisector Theorem
If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
In mathematical notation: If point $P$ is on the perpendicular bisector of $\overline{AB}$, then:
$$PA = PB$$
Converse of the Theorem
If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of that segment.
In mathematical notation: If $PA = PB$, then point $P$ lies on the perpendicular bisector of $\overline{AB}$.
🔑 Putting Both Together:
A point is on the perpendicular bisector of a segment if and only if it is equidistant from the two endpoints. The perpendicular bisector is the locus (set) of ALL points equidistant from $A$ and $B$.