Segments, Length, and Distance

The object and its length

Draw a segment joining $A$ and $B$. The segment is a set of points. Its length is a nonnegative number describing how far apart its endpoints are, measured in a chosen unit.

We write $\overline{AB}$ for the segment and $AB$ for its length. Thus $AB=6$ centimeters describes a measurement; $\overline{AB}$ names the object being measured. Reversing the letters changes neither: $\overline{AB}=\overline{BA}$ as sets of points, and $AB=BA$ as lengths.

Two segments are congruent when their lengths are equal. If $AB=CD$, write $\overline{AB}\cong\overline{CD}$. The segments can occupy different places or point in different directions. Matching tick marks in a diagram declare congruent segments; appearance alone does not.

For distinct endpoints the length is positive. The distance from a point to itself is zero. A coordinate can be negative, but a length cannot.

Distance on a number line

From coordinate $-4$ to coordinate $7$, you travel $4$ units to zero and another $7$ units to the right. The distance is $11$ units. Subtracting the left reading from the right gives that same gap:

Traveling back from $7$ to $-4$ changes the direction but not the distance. The subtraction $-4-7=-11$ records a signed change, so take its absolute value to obtain the length.

We use the number-line distance foundation, often called the ruler postulate: real-number coordinates can be assigned so that each point on the line has exactly one coordinate, each real number is the coordinate of exactly one point on the line, and the distance between two points is the absolute difference of their coordinates. The coordinate scale fixes the unit of length.

For points with coordinates $a$ and $b$:

You may instead subtract the smaller coordinate from the larger. Both methods give a nonnegative distance and work on either side of zero.

Betweenness and segment addition

On the aligned number lines, $7$ lies between $-4$ and $13$. Traveling from $-4$ to $13$ through $7$ covers the whole segment once, with no gap or overlap. The two part lengths therefore add:

The point order introduced in 1.1 can be expressed with coordinates. We accept that coordinates follow this order: for distinct collinear points $A,B,C$ with coordinates $a,b,c$ on one number line, point $B$ is between $A$ and $C$ exactly when $a<b<c$ or $c<b<a$. Either endpoint may come first on the coordinate scale.

We accept the segment addition foundation: if $B$ is between $A$ and $C$, then

The middle letter of the stated order identifies the joining point. A sketch that places a point near the segment is not enough: the betweenness condition includes collinearity.

For four distinct points in order $A,B,C,D$, apply segment addition twice. First, $AC=AB+BC$. Then $AD=AC+CD$, giving $AD=AB+BC+CD$. This joins three adjacent parts without introducing a new addition rule.

Midpoints and bisectors

If a segment is $20$ units long, a point halfway along it leaves $10$ units on each side. Both the location and the equal lengths matter.

A midpoint of $\overline{AB}$ is a point $M$ on the segment such that $AM=MB$. When $A$ and $B$ differ, $M$ cannot be either endpoint: one part would have length zero while the other has the positive length $AB$. Thus $M$ is between them, and segment addition shows that each half has length $AB/2$.

On a number line, endpoints $-4$ and $7$ are $11$ units apart. Half that gap is $5.5$, so the midpoint is $-4+5.5=1.5$. It is also the average of the endpoint coordinates:

For general coordinates $a$ and $b$, move halfway from $a$ toward $b$. The signed change is $b-a$, which may be negative, so half that change gives $a+(b-a)/2=(a+b)/2$. Thus the midpoint’s number-line coordinate is $(a+b)/2$ regardless of endpoint order. This statement concerns coordinates on one line, not a two-coordinate formula in the plane.

A segment bisector is a line, ray, segment, or plane that intersects a segment at its midpoint, sharing just that midpoint with the segment. It cuts the segment into two congruent parts. A bisector need not be perpendicular to the segment. Crossing at a square corner and cutting into equal halves are separate conditions.

Measurement and units

Suppose one endpoint of a strip aligns with the $2.3$ centimeter mark of a ruler and the other with $8.1$ centimeters. The length is the difference, not the final reading:

A ruler measures intervals. The marks at $0$, $1$, $2$, and $3$ enclose three unit intervals, not four. If the zero end is worn, using two other readings and subtracting still measures the gap.

Use the same unit before comparing or adding lengths. Since $1$ centimeter is $10$ millimeters, $5.8$ centimeters is $58$ millimeters. A statement such as $5.8+58=63.8$ cannot combine those measurements as lengths until their units match.

A physical measurement is approximate. If a ruler supports only a nearest-millimeter reading, report that precision rather than invent extra decimal places. An exact length supplied by a geometry problem is different: you may use the stated value exactly. A drawing labeled “not to scale” provides relationships and labels, not a reliable measurement from the page.

Add to the foundations reference

The previous lesson supplied geometric objects and incidence foundations. This lesson adds the rules that connect segments to numbers.

The midpoint results follow from the definitions, segment addition, and algebra. They do not require triangle theorems or facts about angles.

Common mistakes