Segment Addition Postulate Calculator

Enter the coordinates of three collinear points A, B, and C on a number line to find AB, BC, and AC, and check whether the Segment Addition Postulate (AB + BC = AC) holds.

Quick Facts

Segment Addition Postulate
AB + BC = AC
Holds exactly when point B lies between points A and C on the same line.
Distance on a number line
AB = |xB − xA|
Segment length is the absolute value of the difference in coordinates.
Solving for a missing segment
BC = AC − AB
Rearrange the postulate once two of the three lengths are known.

Your Results

Calculated
AB (A to B)
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Distance from A to B
BC (B to C)
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Distance from B to C
AC (A to C)
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Distance from A to C
Postulate Check
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Is AB + BC = AC?

Ready

Enter the coordinates of points A, B, and C, then press Calculate.

How the Segment Addition Postulate Works

The Segment Addition Postulate is a foundational rule in geometry: if point B lies on segment AC and is between points A and C, then the length of AB plus the length of BC equals the length of AC. Written as a formula, that is AB + BC = AC. On a number line, the distance between any two points is the absolute value of the difference of their coordinates, so AB = |xB − xA|, BC = |xC − xB|, and AC = |xC − xA|. This calculator takes the coordinates of three points and both computes those three distances and checks whether the postulate holds — that is, whether B truly sits between A and C.

How the calculation works

Enter the coordinate of each point (A, B, and C) along a single line, plus an optional unit label. The calculator finds AB = |xB − xA|, BC = |xC − xB|, and AC = |xC − xA|. It then compares AB + BC to AC: if they are equal (within rounding), point B is between A and C and the Segment Addition Postulate is satisfied. If AB + BC is greater than AC, B lies outside segment AC — off to one side rather than between the endpoints. This same relationship works in reverse: if you already know two of the three lengths (say AC and AB), you can find the third with BC = AC − AB or AB = AC − BC.

Common mistakes

  • Forgetting the betweenness condition: AB + BC = AC only holds when B is between A and C. If the points are out of order (e.g., B is not on segment AC), the sum will not equal AC.
  • Sign errors: segment lengths are always non-negative, so always take the absolute value of the coordinate difference rather than subtracting in a fixed order.
  • Mixing up segment names: AB, BC, and AC refer to specific pairs of points — double-check which two points define each segment before adding or subtracting.

Real-world applications

  • Proving points are collinear and in a specific order, a common requirement in geometric proofs.
  • Finding an unknown coordinate or unknown segment length when two related lengths are already known, including algebra problems where AB and BC are given as expressions in x.
  • Measuring and laying out evenly spaced points along a straight edge, wall, or fence line, where a middle mark must satisfy AB + BC = AC.
  • Verifying survey or CAD measurements along a straight reference line by checking that partial distances sum to the total distance.

Frequently Asked Questions

What is the Segment Addition Postulate?
The Segment Addition Postulate states that if point B lies between points A and C on the same line, then the length of AB plus the length of BC equals the length of AC: AB + BC = AC.
How do you use the Segment Addition Postulate to find a missing length?
Rearrange AB + BC = AC to isolate the unknown: BC = AC − AB, or AB = AC − BC. For example, if AC = 20 and AB = 8, then BC = 20 − 8 = 12.
How can you tell if point B is between points A and C?
Place the three points on a number line and compute AB = |xB − xA|, BC = |xC − xB|, and AC = |xC − xA|. If AB + BC = AC, then B is between A and C. If the sum does not equal AC, B is not between A and C.
Does the Segment Addition Postulate work with negative coordinates?
Yes. Because distances use absolute value (AB = |xB − xA|), the postulate works the same way whether coordinates are positive, negative, or zero, as long as all three points lie on the same line.

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