Verify or solve for missing segment lengths using the segment addition postulate AB + BC = AC. Supports multiple collinear points with distance tables and visual number lines.
The **segment addition postulate** states that if point B lies between points A and C on a line, then AB + BC = AC. This fundamental axiom of Euclidean geometry is the starting point for virtually every proof involving segment lengths, coordinate geometry, and constructions.
This calculator lets you work with the postulate in four modes: verify that three lengths satisfy AB + BC = AC, or find any one of the three segments when the other two are known. A fifth mode handles **multiple collinear points** — enter any number of coordinates and get every consecutive segment, all-pairs distances, and automatic verification of the postulate for every valid triple. A proportional number-line visualization, step-by-step solution tables, midpoint and ratio cards, and colour-coded verification results make the geometry tangible.
Whether you are a student learning segment relationships, a teacher creating worksheet answer keys, or preparing for a standardized geometry exam, this tool covers every facet of the segment addition postulate with presets, selectable units, and adjustable precision.
The segment addition postulate is one of the first axioms students encounter in a geometry course, yet it appears in nearly every proof and problem set involving segment lengths. This calculator automates the arithmetic so you can focus on understanding the concept: enter known values, and the tool solves for the missing segment, shows each algebraic step, and draws the number line to scale.
The multi-point mode is especially powerful for worksheets and exams that place four or five labelled points on a line and ask for various distances. Instead of computing each pair manually, enter all coordinates once and get a complete distance table and postulate-verification grid. Teachers can use this to generate answer keys instantly.
Segment Addition Postulate: If B is between A and C, then AB + BC = AC. Equivalently, AB = AC − BC or BC = AC − AB. Midpoint of AB = AB / 2 from A.
Result: Verified: 3 + 5 = 8 ✓
Point B lies between A and C. AB + BC = 3 + 5 = 8, which equals AC. The segment addition postulate holds. The midpoint of AB is 1.5 units from A, and AB:BC = 3:5.
In Euclidean geometry, the segment addition postulate (sometimes called the ruler postulate corollary) is invoked whenever you need to relate the whole to its parts. For example, to prove that two segments are congruent, you might show that each equals the same sum of sub-segments. The postulate also underpins the definition of betweenness: B is between A and C if and only if AB + BC = AC. This definition avoids ambiguity and is used in axiomatic treatments of geometry.
On a number line, the segment addition postulate follows directly from the absolute-value distance formula: |AC| = |AB| + |BC| when B is between A and C in terms of coordinate order. In higher dimensions, the postulate generalises to collinear points: three points are collinear if and only if the sum of two distances equals the third. This provides a simple test for collinearity and is used in computational geometry algorithms.
The same additive structure appears elsewhere in geometry. The **angle addition postulate** states that if ray BD lies inside angle ABC, then m∠ABD + m∠DBC = m∠ABC. The **arc addition postulate** applies to circles: adjacent arcs sum to the larger arc. Recognising this common pattern — the "addition postulate" framework — helps unify many geometric arguments under a single logical template.
It states that if B is between A and C on a line segment, then the length AB plus the length BC equals the length AC. It is one of the foundational axioms of Euclidean geometry.
If you know two of the three values (AB, BC, AC), the third is determined: AB = AC − BC, BC = AC − AB, or AC = AB + BC. Select the appropriate mode in the calculator.
Yes. The postulate applies to all real-valued lengths, including decimals and irrational numbers. Enter decimal values directly; set higher precision for more decimal places.
Yes. Switch to "Multiple collinear points" mode and enter any number of coordinates. The calculator computes every consecutive segment, all-pairs distances, and verifies the postulate for every valid triple.
The segment addition postulate is the basis for proving segment congruence, midpoint properties, and many coordinate-geometry results. It lets you replace a whole segment with the sum of its parts (or vice versa) in any equation.
Then B does not lie between A and C on the segment. The points may be collinear with B outside segment AC, or they may not all be collinear at all.