Segment Addition Postulate Calculator
Find a missing segment length, total length, or algebra value using the geometry rule AB + BC = AC when point B lies between A and C.
Segment addition for A, B, and C
The segment addition postulate says that if point B is between point A and point C on the same line, then AB + BC = AC. The two smaller segments add up to the whole segment.
A segment addition postulate calculator calculates missing line segment lengths using the formula AB + BC = AC. The calculator applies the segment addition postulate when point B lies between points A and C. For example, if AC = 15 and AB = 9, then BC equals 6 units.
Use this calculator to find AC from AB and BC, find a missing part from the whole, or solve linear expressions such as AB = 2x + 3, BC = x + 5, and AC = 20.
The diagram updates with the result, so you can see which length is a part and which length is the whole.
Segment result
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AB
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First part
BC
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Second part
AC
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Whole segment
Step-by-step solution
Equation
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Check
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Mode
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How to use the segment addition postulate calculator
- Choose the calculation type: Find the whole segment AC, find a missing part AB or BC, check whether B can be between A and C, split a bisected segment, or solve a linear algebra setup.
- Enter the known values: For number problems, enter the two known segment lengths. For algebra, enter each expression as ax + b.
- Check the diagram: Point B must be between A and C, so AB and BC are the two smaller pieces of AC.
- Read the steps: The result shows the formula, substitution, arithmetic, and final answer.
- Verify the answer: A correct result must satisfy AB + BC = AC.
Which calculator mode should you use?
| Mode | Use it when | Inputs needed | Output |
|---|---|---|---|
| Find AC | You know both adjacent partial lengths. | AB and BC | Total length AC |
| Find AB or BC | You know the whole and one part. | AC plus the known part | Missing segment length |
| Check B between A and C | You need to test whether a diagram or measurement set is consistent. | AB, BC, and AC | Equal / not equal conclusion |
| Segment bisector | B is a midpoint or bisector of AC. | AC | AB and BC as equal halves |
| Linear expressions | A homework problem gives expressions with x. | AB, BC, AC as ax + b | x plus all three segment lengths |
The strongest workflow is to solve first, then use the check line to verify that the final answer still satisfies part + part = whole.
Segment addition postulate formula
The postulate is used when three points are collinear and point B lies between A and C. In that case, the distance from A to B plus the distance from B to C equals the distance from A to C.
AB + BC = AC
Missing part = whole - known part
For algebra: expression AB + expression BC = expression AC
The rule is simple, but it prevents a common geometry mistake: adding segments that are not adjacent parts of the same straight segment.
Formula reference: Texas Instruments Education - Segment Addition Postulate.
Worked examples
| Problem | Setup | Work | Answer |
|---|---|---|---|
| Find AC | AB = 8, BC = 5 | AC = 8 + 5 | AC = 13 |
| Find AB | AC = 24, BC = 9 | AB = 24 - 9 | AB = 15 |
| Find BC | AC = 31, AB = 12 | BC = 31 - 12 | BC = 19 |
| Solve for x | AB = 2x + 3, BC = x + 5, AC = 20 | 2x + 3 + x + 5 = 20 | x = 4, AB = 11, BC = 9 |
Coordinate and number line segment addition
On a number line or coordinate axis, segment length is distance, not just subtraction in one direction. If A, B, and C are points on the same line and B is between A and C, use absolute value to find each part, then check that the partial lengths add to the total length.
AB = |xB - xA|
BC = |xC - xB|
AC = |xC - xA|
Example: if A = -3, B = 2, and C = 8, then AB = |2 - (-3)| = 5, BC = |8 - 2| = 6, and AC = |8 - (-3)| = 11. Since 5 + 6 = 11, point B is between A and C and the segment addition postulate works.
Coordinate segment reference: Math Open Reference - Line Segment.
Proof and homework wording template
Many geometry assignments ask for a reason, not only a number. Use a short proof format when the problem asks you to justify a missing value, write a statement-reason proof, or explain why two partial lengths make one whole segment.
Given
State the point order and measurements: B is between A and C, AB = 9, and AC = 15.
Equation
Write the postulate: AB + BC = AC. Substitute known values: 9 + BC = 15.
Conclusion
Solve and label the answer: BC = 6 units by the segment addition postulate.
Copy-ready sentence: Since B is between A and C, AB + BC = AC by the segment addition postulate. Substituting the known lengths gives the missing segment.
Proof reference: CourseNotes - Segment Addition Postulate.
Point order reference table
| Point order | Middle point | Correct equation | Meaning |
|---|---|---|---|
| A - B - C | B | AB + BC = AC | The calculator's default diagram. |
| A - C - B | C | AC + CB = AB | C is between A and B, so AB is the whole. |
| B - A - C | A | BA + AC = BC | A is between B and C, so BC is the whole. |
| Not collinear | None | Do not use segment addition | Use a distance, triangle, or coordinate method instead. |
Before solving, identify the middle point. The two shorter pieces must share that middle point and sit on the same straight line.
What to check before using the answer
The formula only works when B lies between A and C on the same straight segment. If the diagram places the points in a different order, rename the segments before calculating.
Point order
AB + BC = AC assumes the order is A, then B, then C. If the order is A-C-B, the equation changes.
Positive lengths
A segment length cannot be negative. If a missing part is negative, the known values do not describe a valid segment.
Algebra check
After solving for x, substitute back into AB, BC, and AC. The check should read part plus part equals whole.
Interesting fact
Segment addition is modern classroom wording, but its logic sits inside the much older tradition of Euclidean geometry. Encyclopaedia Britannica notes that Book I of Euclid's Elements began with 23 definitions, 5 postulates, and 5 common notions. Those foundations include ideas about adding equals and comparing a whole with a part. A segment equation such as AB + BC = AC is a simple example of that same part-whole reasoning.
Frequently Asked Questions
What is the segment addition postulate in geometry?
The segment addition postulate is a geometry rule about collinear points on the same line segment. If point B is between endpoint A and endpoint C, then the partial length AB plus the partial length BC equals the total length AC. In formula form, the equation is AB + BC = AC, and the calculator uses that formula to turn the diagram into a clear solution.
How do I find a missing segment length?
If you know both smaller segment measurements, add them to find the whole distance. If you know the total length and one partial length, subtract the known segment from the whole to find the missing value. For example, if AC = 18 and AB = 7, then BC = 18 - 7 = 11. The answer should always make the check AB + BC = AC true.
Can the calculator solve a segment expression with a variable?
Yes. Choose the linear expressions mode and enter AB, BC, and AC as ax + b expressions. The calculator builds the segment addition postulate equation AB + BC = AC, solves for the variable x, and then substitutes x back into each segment length. This is useful for geometry homework where the diagram gives expressions instead of direct measurements.
Why does point B need to be between endpoints A and C?
The equation depends on AB and BC being adjacent parts of the same whole line segment. If B is not between A and C, then AB and BC do not combine to form AC, even if the three points appear on the same line. In a coordinate diagram, check the point order first; the distance formula may give correct lengths, but the postulate only applies when the partial lengths sit next to each other.
What if the missing value is negative?
A negative segment length means the entered values do not describe a valid A-B-C diagram. For example, AC = 10 and AB = 14 cannot produce a positive BC when B is between A and C, because one part is already longer than the whole segment. In a proof or homework problem, a negative result usually means the endpoints were labeled in the wrong order, the equation was copied incorrectly, or the given measurement is inconsistent.
How does the calculator check whether B is between A and C?
The check mode compares AB + BC with AC. If the two sides of the equation are equal within the displayed rounding tolerance, then the measurements are consistent with B lying between A and C on a straight segment. If AB + BC does not equal AC, then those three lengths cannot describe that A-B-C point order.
What does the segment bisector mode do?
The segment bisector mode assumes point B divides AC into two congruent parts. Enter the total length AC, and the calculator divides it by 2 to get AB and BC. For example, if AC = 20 units and B bisects AC, then AB = 10 units and BC = 10 units.
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Disclaimer: This segment addition postulate calculator is for general educational and informational use only. It is designed to support geometry practice, homework checking, and algebra review. It may not match every teacher's notation, textbook convention, diagram label, proof requirement, or assessment format. Always confirm the point order, units, and classroom instructions before using the final answer.
Last updated: May 19, 2026