Compare, add, subtract, and split — the same moves you made with numbers, now with length.
Point 4 of 5 in this lesson: 13.4.4 The midpoint
A segment is a finite straight piece with two endpoints — and unlike a full line, it carries one number with it: its length. The moment a shape has a number attached, every bit of arithmetic you already own comes roaring back to life. You can compare two segments, copy one exactly, add them end to end, subtract a piece, cut a segment cleanly in half, or chop it into any number of equal parts. The point line surface solid story of the last three lessons gave us the segment; this lesson teaches us to measure and compute with it.
How do you tell which of two segments is longer? Two honest ways, and you'll use both for the rest of your life.
Measure both. Lay each segment along a ruler, read off the two lengths in the same unit, and compare the numbers. AB = 6 cm and CD = 4 cm, so AB > CD. Length is just a number, and numbers know how to be compared.
Or slide one onto the other. Put endpoint A exactly on endpoint C and lay the two segments along the same direction. Now only the far ends matter: whichever segment's far end reaches further is the longer one. This trick needs no ruler marks at all — it's the same idea your compass will use in the next section.
Comparing length is comparing numbers. Align one pair of endpoints, then read the other end — or just measure. Either way, exactly one of AB > CD, AB = CD, AB < CD holds.
Suppose you want a second segment exactly as long as AB, but you're not allowed to read any ruler marks. The compass does it perfectly. Open the compass so its two points sit on A and B — now the compass is the length AB, frozen. Draw a fresh ray starting at a new point A′, put the compass point on A′, and swing an arc that crosses the ray. Call the crossing point B′. Because you never changed the opening, A′B′ = AB.
A single hatch tick ′ across two segments says they have the same length. A copied segment always earns the same tick as its original, because the compass opening never changed. We'll use this tick-mark language constantly in geometry.
Lay one segment right after another, head to tail along a straight line, and their lengths simply add. If B sits between A and C on a straight path, then
AC = AB + BC.
Read it backwards and you get subtraction: chop the piece BC off the whole and what's left is AB = AC − BC. This is exactly the "part + part = whole" you met with numbers — only now the parts are lengths laid end to end.
The clean rule AC = AB + BC needs B to lie between A and C on the line. If B were off to the side, the three lengths would not add up like this.
There is one special point on a segment that splits it into two equal halves. It's called the midpoint. The point M is the midpoint of AB when
AM = MB = 12AB.
Slide M below and find the one spot where the left piece and the right piece are truly equal. Everywhere else, one half is longer than the other — only at the midpoint do the two ticks finally match.
The midpoint is the segment's exact center: AM = MB = ½AB. So if you know the whole, each half is half of it; and if you know a half, the whole is twice as long. (Soon you'll meet the angle's version of this point — the bisector — in lesson 13.6.)
The midpoint cut a segment into two equal pieces. Why stop at two? Place two equally spaced cuts and you get three equal parts (the cut points are the trisection points); three cuts give four equal parts; and in general k equal pieces need k − 1 interior points. Each piece is just
each part = AB ÷ k.
It's division, drawn out on a line. To split a 12-unit segment into 3 equal parts, each part is 12 ÷ 3 = 4 units. (The compass-and-straightedge way of making these cuts perfectly without a ruler comes a little later; here we just see what equal division means.)
A segment carries a single number — its length — and that number unlocks all of arithmetic:
| Move | What it says |
|---|---|
| Compare | align one end, or measure: AB > = < CD |
| Copy | compass gives A′B′ = AB (same tick) |
| Add | AC = AB + BC (B between A and C) |
| Subtract | AB = AC − BC |
| Halve | midpoint M: AM = MB = ½AB |
| Divide | k equal parts, each = AB ÷ k |
Keep this table in your pocket. In lesson 13.6 you'll watch every single move come back — compare, add, subtract, bisect — this time for angles.
PQ = 9 cm and RS = 9 cm. Compare the two segments.
PQ = RS — they have the same length, so the segments are equal.
On a straight line, B lies between A and C. If AC = 12 and AB = 7, find BC.
BC = AC − AB = 12 − 7 = 5.
M is the midpoint of AB, and AB = 14 cm. Find AM.
AM = ½ AB = ½ × 14 = 7 cm (and MB = 7 cm too).
You copy segment AB onto a ray with a compass, never changing its opening, getting A′B′. What is the relationship between A′B′ and AB, and why?
A′B′ = AB. The compass opening was set to the exact length AB and never changed, so the copied length must equal the original — that's why both get the same tick mark.
A 15 cm segment is divided into 3 equal parts. How long is each part?
Each part = 15 ÷ 3 = 5 cm. (The two cut points are the trisection points.)
Point P lies on AB with AP = ⅓ AB. Where does P sit, and how does PB compare to AP?
P is the trisection point nearer A — one third of the way from A to B. Then PB = ⅔ AB, so PB is twice as long as AP.
Six questions to lock it in. Tap the answer you think is right.
A segment is the first geometric object with a number attached, and that number behaves exactly like the numbers students already know. So this lesson is really arithmetic in disguise: comparing is comparing numbers, "AC = AB + BC" is addition of parts, the midpoint is halving, and equal division is plain division. Naming these moves now makes the angle version in 13.5–13.6 feel familiar instead of new.
"AC = AB + BC" is only true when B lies between A and C on the line — the betweenness is doing real work. Off the line, the three lengths form a triangle and the equation fails (in fact the two short sides exceed the long one). It's worth drawing that case once so the "+" doesn't get treated as automatic.
Common Core: builds the measurement-and-construction groundwork behind 7.G.A.2 (drawing geometric figures, including with ruler and compass) and the high-school congruence strand HSG-CO.D.12 (copying a segment, bisecting a segment with compass and straightedge). The midpoint and equal-division ideas also support coordinate-geometry work in later grades.