Segment Addition & Midpoints
Adding lengths along a line, and finding the exact middle.
The explanation
If a point sits between two others, the two short pieces add up to the whole. With B between A and C:
AB + BC = AC
That is the Segment Addition Postulate, and it turns a diagram into an equation. If AB = 3x, BC = 2x + 1 and AC = 26, then 3x + 2x + 1 = 26, so x = 5.
A midpoint cuts a segment into two equal halves. If M is the midpoint of AB, then AM = MB, and each is half of AB.
A segment bisector is anything — a line, ray or segment — that passes through the midpoint. A perpendicular bisector does both: it passes through the midpoint and meets the segment at 90°.
The Segment Addition Postulate states that for B between A and C, AB + BC = AC. Betweenness is a genuine hypothesis, not a formality: the relation fails if B lies outside the segment, which is why diagrams must be read for order before the equation is written.
A midpoint M of AB satisfies AM = MB = ½AB, and it is unique. On the coordinate plane this becomes the midpoint formula, the componentwise average ((x₁+x₂)/2, (y₁+y₂)/2).
The perpendicular bisector has a characterisation worth remembering because it does most of the work in later constructions and proofs: a point lies on the perpendicular bisector of a segment exactly when it is equidistant from the two endpoints. That equidistance property is what makes the perpendicular bisectors of a triangle's sides meet at the circumcentre, the centre of the circle through all three vertices.
The parallel result for angles is the angle bisector, whose points are equidistant from the two sides of the angle.
Worked example
M is the midpoint of AB. AM = 4x − 3 and MB = 2x + 9. Find AB.
- Midpoint means the halves are equal: 4x − 3 = 2x + 9.
- 2x = 12, so x = 6.
- AM = 4(6) − 3 = 21.
- AB is twice one half.
Answer: AB = 42
Common mistakes
- Solving for x and reporting it as the length when the question asked for AB.
- Applying segment addition without checking which point is actually between the others.