Segment Addition Postulate Calculator
Enter any two of the three segment lengths (AB, BC, or AC) and the calculator finds the third instantly using the segment addition postulate: AB + BC = AC. You can type plain numbers or algebraic expressions such as "2x+3" or "5x-1" and the tool solves for x, then evaluates each length. The step-by-step panel shows the full working, and a live diagram updates to match your result.
What is the segment addition postulate?
The segment addition postulate is a foundational rule in Euclidean geometry: if point B lies between points A and C on a line, then the lengths of the two smaller segments sum to the length of the whole segment, expressed as AB + BC = AC. This postulate is used to set up equations, verify collinearity (whether three points are on the same line), find missing lengths, and work with algebraic expressions involving segment lengths. It applies only when B is strictly between A and C, meaning all three points must be collinear and B must not coincide with either endpoint.
How to solve segment addition problems with algebra
When segment lengths are given as algebraic expressions (for example AB = 2x + 3 and BC = x - 1 and AC = 14), you substitute into AB + BC = AC, combine like terms, and solve the resulting linear equation for x. Once x is found you substitute back to evaluate each expression and obtain the numerical lengths. This calculator handles that process automatically: enter each expression in the input field and the tool parses it, forms the equation, solves for x, and displays every length. Always verify at the end that AB + BC equals AC with the numerical values.
Special cases: midpoint and golden ratio
A segment bisector is a point (or line, ray, or plane) that cuts a segment into two equal halves. If AB = BC then B is the midpoint of AC, and AC = 2 * AB = 2 * BC. You can check for a midpoint by looking at the ratio shown below the result: if it equals 1, B bisects AC. Another notable special case is the golden ratio (approximately 1.618): when the longer sub-segment divided by the shorter equals about 1.618, the two parts form a golden proportion. This ratio appears in classical architecture, art, and nature, and this calculator flags it automatically when your inputs produce it.
Unit choices and real-world applications
The postulate is dimensionless: it works equally well in centimetres, metres, inches, or feet, or with unitless abstract lengths. Select your preferred unit from the dropdown and it will appear beside every result and in the step-by-step panel. In practice the segment addition postulate is used in architectural drawings to split a wall into sections, in engineering to calculate spans and offsets, in navigation to combine measured distances, and in any technical drawing where partial measurements must add to a known total.
Segment addition postulate - common problem types
| Given | Find | Formula | Example |
|---|---|---|---|
| AB and BC | AC | AC = AB + BC | AB = 6, BC = 10 → AC = 16 |
| BC and AC | AB | AB = AC - BC | BC = 9, AC = 15 → AB = 6 |
| AB and AC | BC | BC = AC - AB | AB = 5, AC = 14 → BC = 9 |
| Expressions with x | x, then each length | Set up and solve a linear equation | AB = x+2, BC = 3x, AC = 14 → x = 3 |
A quick reference for the three rearrangements of AB + BC = AC and when each is used.
Frequently asked questions
What is the segment addition postulate formula?
The formula is AB + BC = AC, where B is a point that lies between A and C on the same straight line. You can rearrange it to AB = AC - BC (to find the first segment) or BC = AC - AB (to find the second segment) depending on which value is unknown.
Can I enter expressions like "2x + 3" instead of a number?
Yes. Type any simple linear expression such as "2x+3", "x-1", "5x", or "-3x+7" directly into the input field. The calculator parses the expression, forms the equation AB + BC = AC with the substituted expressions, solves for x, and then evaluates and displays all three segment lengths.
What does "point B lies between A and C" mean?
It means A, B, and C are three collinear points (they all sit on the same straight line) and B is not at either endpoint. Imagine a number line: if A is at position 0 and C is at position 16, then any point B at a position strictly between 0 and 16 satisfies "B lies between A and C." If B is outside that range, the segment addition postulate does not apply.
How do I check if three points are collinear using this postulate?
Measure or calculate AB, BC, and AC independently. If AB + BC equals AC exactly, the three points are collinear and B lies between A and C. If AB + BC is greater than AC or less than AC, the points are not collinear, or B does not lie between A and C.
What if AB and BC are equal?
When AB equals BC, point B is the midpoint of segment AC and is called a segment bisector. In that case AC = 2 * AB = 2 * BC. This calculator shows a ratio of 1.000 in the output and the insight panel will note that B is the midpoint.
Does the postulate work with more than three points?
Yes. The segment addition postulate extends to any number of collinear points. For four points A, B, C, D in order on a line: AB + BC + CD = AD. You can chain as many segments as needed, and the total of all sub-segments equals the full length between the two outermost points.