Between

Between

Point B is between points A and C if it is on the line segment connecting A and C.

A line segment with point A on the left, point B in the middle, and point C on the right, showing B between A and C.

See also

Key Formula

AB + BC = AC

Where:

$A, B, C$ = Three collinear points where B is between A and C
$AB$ = The distance from point A to point B
$BC$ = The distance from point B to point C
$AC$ = The total distance from point A to point C

Worked Example

Problem: Points A, B, and C are collinear. If AB = 5, BC = 8, and AC = 13, is B between A and C?

Step 1: Write the betweenness condition: B is between A and C if AB + BC = AC.

AB + BC = AC

Step 2: Substitute the given distances into the left side of the equation.

5 + 8 = 13

Step 3: Compare with AC. Since 13 = 13, the equation holds true.

13 = 13 \quad \checkmark

Answer: Yes, B is between A and C because AB + BC = AC.

Another Example

This example uses algebra to find unknown segment lengths, which is the most common way betweenness appears in geometry homework and tests.

Problem: Point B is between A and C. If AB = 3x + 2 and BC = x + 6, and AC = 24, find x and the lengths AB and BC.

Step 1: Since B is between A and C, apply the segment addition postulate.

AB + BC = AC

Step 2: Substitute the algebraic expressions and the total length.

(3x + 2) + (x + 6) = 24

Step 3: Combine like terms and solve for x.

4x + 8 = 24 \implies 4x = 16 \implies x = 4

Step 4: Substitute x = 4 back into each expression to find the individual lengths.

AB = 3(4) + 2 = 14, \quad BC = 4 + 6 = 10

Step 5: Verify: 14 + 10 = 24, which equals AC.

14 + 10 = 24 \quad \checkmark

Answer: x = 4, so AB = 14 and BC = 10.

Frequently Asked Questions

What is the Segment Addition Postulate and how does it relate to betweenness?

The Segment Addition Postulate states that if B is between A and C, then AB + BC = AC. This postulate is essentially the formal geometric statement of betweenness. Whenever you use the concept of between in geometry proofs or problems, you are invoking this postulate.

Can a point be 'between' two other points if all three are not on the same line?

No. For B to be between A and C, all three points must be collinear — that is, they must lie on the same straight line. If the three points form a triangle or are otherwise non-collinear, then no point is between the other two in the geometric sense, because AB + BC would be strictly greater than AC by the triangle inequality.

Does 'between' include the endpoints?

In standard geometry, between is exclusive of the endpoints. If B is between A and C, then B is a distinct point that is neither A nor C. This is similar to the distinction between an open interval (exclusive) and a closed interval (inclusive) on a number line.

Between (exclusive) vs. Between (inclusive)

	Between (exclusive)	Between (inclusive)
Definition	B is strictly interior to segment AC; B ≠ A and B ≠ C	B can be any point on segment AC, including the endpoints A or C
Notation (number line)	Open interval: (a, c) or a < x < c	Closed interval: [a, c] or a ≤ x ≤ c
When to use	Standard geometry betweenness; when endpoints are excluded	When a range of values includes its boundary values

Why It Matters

Betweenness is foundational to geometry proofs — the Segment Addition Postulate (AB + BC = AC) appears in nearly every unit on segments, midpoints, and coordinate geometry. You will also encounter betweenness on number lines in algebra when describing intervals and inequalities. Standardized tests like the SAT and ACT frequently ask you to find unknown lengths using the betweenness relationship.

Common Mistakes

Mistake: Assuming B is between A and C just because all three points are collinear.

Correction: Collinearity alone is not enough. You must verify that AB + BC = AC. If instead AC + CB = AB, then C is between A and B, not B between A and C. Always check which point lies in the interior.

Mistake: Confusing 'between' with 'midpoint' and assuming the between point divides the segment into two equal parts.

Correction: A point between two others can be anywhere along the interior of the segment, not necessarily at its center. A midpoint is a special case of betweenness where AB = BC.

Related Terms

Point — The fundamental object positioned between others
Line Segment — The segment on which a between point lies
Exclusive — Between typically excludes the endpoints
Inclusive — Inclusive between includes the endpoints
Collinear — All three points must be collinear
Midpoint — Special case where the between point bisects the segment
Segment Addition Postulate — Formal postulate expressing betweenness as AB + BC = AC