Point, line, ray, and segment
The handful of starting words every later lesson leans on, and how to check whether a point really sits between two others.
Every lesson from here on leans on a handful of words: point, line, ray, segment, and between. They are agreed rather than proved — there is nothing smaller to build them from — so this lesson is short on purpose: there isn’t much to say, only a few distinctions worth getting right the first time.
A point is a place with no size of its own. A line runs straight through two points and never stops, in either direction. A ray also never stops, but only in one direction — it starts at a point and runs straight from there. A segment stops at both ends: it is the piece of a line between two of its points, together with those two points themselves.
Four rows, top to bottom: a single point, P; a line through A and B, running off both edges of the frame; a ray from O, running off one edge with an arrowhead; and a segment from C to D, stopping at both ends. This needs JavaScript switched on.
“Straight” is doing real work in each of those definitions, and it means exactly one thing: of every path between two points, a straight one is the shortest. Nothing else about it is proved here — the same way “point” is not proved, only agreed on — because there is nowhere shorter to start from.
A segment has a length: the distance from one end to the other, a single number that is never negative and does not care which end you start counting from. The point sitting exactly that same distance from each end is the segment’s midpoint — not approximately in the middle, but the one specific point where the two halves are equal.
That equal-halves idea is easy to state and surprisingly easy to get wrong by eye. Three points can look centred in a drawing while sitting nowhere near a straight line at all — and even once they genuinely are on one line, whether one sits between the other two only has an answer once you check it, not once it looks right.
Two points, A and C, fixing a line, and a third point, B, free to sit anywhere. A live reading says whether B is on the line, and if so, which of the three regions it is in — and once it is between A and C, how the segment splits, noting when the two pieces are equal. This needs JavaScript switched on.
The moment the two pieces match, B has also found the segment’s midpoint — the one place on the whole line where this happens.
What decides whether B is between A and C is never how the picture looks. It is arithmetic: if AB and BC add up to exactly AC, with nothing left over and nothing overlapping, B is between A and C.
And a midpoint is the special case of that same equation where the two pieces are also equal to each other:
AB is 4 units and BC is 6 units, with B between A and C. What is AC?
.
AB is 14 units. M is the midpoint of AB. What is AM?
Half of 14 is 7, so .
A segment is 7 units long. Its midpoint does not land on a whole grid square — is that a problem?
No. A midpoint is exactly one point, whether or not it happens to fall on a square you can count to: half of 7 is 3.5, and the midpoint sits exactly there, 3.5 units from each end.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.