Segment Addition Postulate Calculator

What Is the Segment Addition Postulate?

The Segment Addition Postulate is a fundamental axiom in Euclidean geometry. It states that if three points $A$, $B$, and $C$ are collinear (lying on the same straight line) and point $B$ lies directly between point $A$ and point $C$, then the sum of the lengths of the two smaller segments equals the total length of the combined segment:

$$AB + BC = AC$$

Here, $AB$ denotes the distance from $A$ to $B$, $BC$ denotes the distance from $B$ to $C$, and $AC$ is the overall distance between endpoints $A$ and $C$.

Collinearity and the Betweenness Condition

For the Segment Addition Postulate to hold, two conditions must be satisfied:

• Collinearity: All three points $A$, $B$, and $C$ must lie on the same 1D line. If the points form a triangle instead of a straight line, the Triangle Inequality applies ($AB + BC > AC$).
• Betweenness: Point $B$ must lie physically between $A$ and $C$. If $C$ is between $A$ and $B$, the relation becomes $AC + CB = AB$.

To determine distances between coordinates in the Cartesian plane before applying this postulate, use our 2D Distance Calculator.

Formulas Derived from the Postulate

Depending on which segment length is unknown, the postulate can be rearranged algebraically:

• Finding the Total Segment $AC$: $$AC = AB + BC$$
• Finding Subsegment $AB$: $$AB = AC - BC$$
• Finding Subsegment $BC$: $$BC = AC - AB$$

Algebraic Segment Addition Problems

In middle school and high school geometry curricula, segment lengths are often given as linear algebraic expressions in terms of an unknown variable $x$. For example:

$$AB = 2x + 3, \quad BC = 4x - 1, \quad AC = 32$$

Applying the postulate gives:

$$(2x + 3) + (4x - 1) = 32$$ $$6x + 2 = 32 \implies 6x = 30 \implies x = 5$$

Substituting $x = 5$ back into the segment expressions yields:

$$AB = 2(5) + 3 = 13, \quad BC = 4(5) - 1 = 19, \quad AC = 13 + 19 = 32$$

Special Case: The Midpoint of a Segment

When point $B$ is the exact midpoint of segment $AC$, the two subsegments are congruent ($AB = BC$):

$$AB = BC = \frac{1}{2}AC \quad \text{and} \quad AC = 2 \cdot AB = 2 \cdot BC$$

To compute coordinate midpoints or divide segments into custom proportions, explore our Midpoint Calculator and Ratios of Directed Line Segments Calculator. You can also analyze triangle segments with our Midsegment of a Triangle Calculator.