# Segment Bisector: Definition, Types & Examples

## What Is a Segment Bisector?
A **segment bisector** is a point, line, ray, or line segment that passes through the **midpoint** of a given segment and divides it into **two congruent parts**, meaning two parts of equal length. The defining feature is the midpoint: a bisector must hit it, and any object that does is a bisector, regardless of the angle it makes.

If segment AB has midpoint M, then a bisector is anything that passes through M so that AM = MB. The two halves are **congruent segments**, written \( \overline{AM} \cong \overline{MB} \). A single segment can have **infinitely many** bisectors, because infinitely many lines can pass through one point, fanning out in every direction.

## The Four Types of Segment Bisector
A bisector is named by _what it is_, not by what it does. The job is always the same, but the object doing the job can be any of four things.

| Bisector | What it is | Note |
| --- | --- | --- |
| **Point** | A single marked midpoint M | The minimal bisector, just the dividing point itself |
| **Line** | A full line through M | Runs both ways forever, crossing at M |
| **Ray** | A ray with its path through M | Starts at one end, runs through M |
| **Line segment** | A segment that crosses at M | Finite, but still passes through the midpoint |

The fifth case, a **plane** cutting a segment at its midpoint, only matters once the segment sits in 3D space.

## The Perpendicular Bisector: the Special 90° Case
A **perpendicular bisector** is a bisector that crosses the segment at a **right angle** (90°), so it both passes through the midpoint _and_ meets the segment perpendicularly. Every segment has infinitely many bisectors but exactly **one** perpendicular bisector, because there is only one line through the midpoint at a right angle.

A segment bisector only has to pass through the midpoint; the angle can be anything. Perpendicular is the one case where that angle happens to be exactly 90°. For the deeper treatment of that special case, including its equidistance property, see the [perpendicular bisector](/content/math/geometry/perpendicular-bisectors/index.html) article.

## How to Find a Segment Bisector Using the Midpoint Formula
Because a bisector is defined by the midpoint, finding one comes down to finding that point. When the endpoints are given as coordinates \((x_1,y_1)\) and \((x_2,y_2)\), the **midpoint formula** locates the bisecting point directly:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Each coordinate of the midpoint is the **average** of the two endpoints' matching coordinates, which is exactly what "halfway between" means. Any line, ray, or segment you draw through this M is a bisector.

## Examples of the Segment Bisector

### **Example 1** - Ray \( \overrightarrow{ST} \) bisects segment \( \overline{AB} \), which is 20 cm long. How long is each half?
A bisector splits the segment into two equal parts, so each half is \( \frac{20}{2} = 10 \) cm.

Final answer: each half is 10 cm.

### **Example 2** - Find the midpoint of the segment from A(−3,4) to B(5,−2)  
Using the midpoint formula,
\[ M = \left( \frac{-3 + 5}{2}, \frac{4 + (-2)}{2} \right) = (1, -3) \]
Final answer: M = (1,1).

### **Example 3** - The midpoint of \( \overline{PQ} \) is M(4,−2) and one endpoint is P(3,h) with Q(7,7). Find h.
The y-coordinate of the midpoint is the average of the endpoints' y-values,
\[ -2 = \frac{h + 7}{2} \Rightarrow -4 = h + 7 \Rightarrow h = -11 \]
Final answer: h = -11.

### **Example 4** - A segment runs from C(2,1) to D(8,9). Find the point where a bisector must cross
\[ M = \left( \frac{2 + 8}{2}, \frac{1 + 9}{2} \right) = (5, 5) \]
Final answer: the bisector crosses at M=(5,5).

### **Example 5** - Two students draw bisectors of the same segment \( \overline{AB} \). Are both correct?
Both lines are correct. A bisector only has to pass through the midpoint, and both lines do.
Final answer: yes, both are valid segment bisectors.

### **Example 6** - The midpoint of \( \overline{AB} \) is M(3,5) and one endpoint is A(1,2). Find the other endpoint B.
Using the midpoint formula backwards,
\[ B = (2 \cdot 3 - 1, 2 \cdot 5 - 2) = (5,8) \]
Final answer: B = (5,8).

## Where Segment Bisectors Earn Their Keep

- **Triangle and shape construction.**
- **Engineering and CAD.**
- **Surveying and land division.**
- **Computer graphics.**

## Where Students Trip Up on Segment Bisectors

### **Mistake 1:** Assuming every bisector is perpendicular

### **Mistake 2:** Subtracting in the midpoint formula

### **Mistake 3:** Forgetting a segment can have many bisectors

## Key Takeaways

- A **segment bisector** is a point, line, ray, or segment that divides it into two congruent halves.
- The four types are point, line, ray, and segment; a plane is added only for 3D work.
- A segment has infinitely many bisectors but only one perpendicular bisector.
- The midpoint formula finds the bisecting point by averaging the endpoints.
- Common mistakes include confusing perpendicular bisectors with segment bisectors and improper use of the midpoint formula.