What Is a Midpoint?

How to prove a point is halfway

A midpoint belongs on the segment it divides. If segment AB has midpoint M, then M lies between A and B, and the two parts are AM and MB. The important fact is AM = MB. The letters and the equal-length statement give us a way to check the idea instead of relying on our eyes.

You can picture a midpoint by folding a paper strip so its two ends line up. The fold mark shows the halfway place. A segment on a screen does not physically fold, of course, but the same balance idea is present. If one side is even a little longer, the point is not the midpoint.

Midpoints appear in many later lessons. The midpoint of a side helps define a triangle's median. The midpoint formula finds the halfway coordinate between two points. A segment bisector passes through a midpoint, and symmetry often places important points at the middle of a figure.

Be careful when a diagram is not drawn to scale. A point may look centered because of the way the picture was made, but the written measurements or coordinates are the real evidence. Check both pieces, use the formula when needed, and remember that halfway means equal distances on both sides.

• The midpoint divides a segment into two equal parts.
• A midpoint must lie on the segment it is dividing.
• The two resulting segments are congruent.
• Only a segment can have a midpoint in the usual Euclidean sense.

Midpoints in diagrams and coordinates

• Use midpoint when splitting a segment into two equal lengths.
• Use it in coordinate geometry through the midpoint formula.
• Use it in triangles when defining medians and midsegments.

Three midpoint errors to catch

• Do not place the point near the center and assume that is enough without equal lengths.
• Do not call a point a midpoint if it is off the segment.
• Do not talk about the midpoint of a whole line or a ray in the same way you do for a segment.