Segment Addition Postulate Calculator (solve AB + BC = AC for x)

Understanding the segment addition postulate calculator — concept and principle

This calculator solves segment addition problems: when point B lies between A and C, AB + BC = AC. Given each length as an expression in x, it finds x and then the lengths of AB, BC and AC, and checks that every length is positive. Use it for geometry homework on segments, midpoints and algebraic lengths.

The two parts add up to the whole: x = 4 makes AB = BC = 11, so B is the midpoint.

How it is calculated

With AB = a₁x + b₁, BC = a₂x + b₂ and AC = a₃x + b₃:

Write the postulate: (a₁x + b₁) + (a₂x + b₂) = a₃x + b₃.
Collect x terms: (a₁ + a₂ − a₃)·x = b₃ − b₁ − b₂.
Divide to get x.
Substitute x into each expression.

A coefficient of 0 makes that length a plain number. Every length must come out positive, or the expressions do not describe a real segment.

The principle

The segment addition postulate is a basic assumption of Euclidean geometry: if B is on segment AC between A and C, walking from A to B and then from B to C covers exactly the distance from A to C. The whole equals the sum of its parts. When the lengths are written with a variable, the postulate turns into a linear equation, so solving it is ordinary algebra. If B is the midpoint, AB = BC as well, which gives a second equation.

Worked example

Default inputs: AB = 2x + 3, BC = 3x − 1, AC = 22.

(2x + 3) + (3x − 1) = 22.
5x + 2 = 22, so 5x = 20.
x = 4.
AB = 2(4) + 3 = 11; BC = 3(4) − 1 = 11; AC = 11 + 11 = 22.
AB is 50% of AC, so B is the midpoint of AC in this example.

Each point is the calculator re-run with only “AC — constant b₃” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

AC — constant b₃	Value of x
5.5	0.7
11	1.8
16.5	2.9
22	4
27.5	5.1
33	6.2
44	8.4

How to read the result

x is only a helper; the lengths are usually what the question asks for. Check that AB + BC equals AC and that all three are positive. The share of AC tells you where B sits: 50% means the midpoint, less than 50% means B is closer to A.

Common mistakes

Setting AB + AC = BC; the longest segment, AC, is the sum of the other two.
Forgetting to substitute x back to get the lengths.
Sign errors when moving constants across the equal sign.
Accepting a negative length instead of rechecking the expressions.

Keep in mind: The postulate applies only when B lies on segment AC between A and C. If B is not between them, or not on the same line, AB + BC is greater than AC.

Frequently asked questions

What is the segment addition postulate?

If B is between A and C on a line segment, then AB + BC = AC.

How is it used with midpoints?

A midpoint makes AB = BC, so each half is AC ÷ 2.

What if I get a negative length?

The given expressions cannot all be correct for a real segment; check the problem statement.

Sources: Euclidean geometry: segment addition postulate (AB + BC = AC when B lies between A and C)

Checked 9 Nov 2026. Standard segment addition postulate of Euclidean geometry (if B is between A and C, then AB + BC = AC), solved here as a linear equation in x.

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