Points, lines and rays

Geom­e­try is the study of shapes and space. Every shape you will ever draw, a tri­an­gle, a square or a cir­cle, is built from a few sim­ple ideas: points, lines, line seg­ments and rays. In this les­son you will learn what each one is, how to name it and how to write it. These words come up in every geom­e­try les­son after this one, so learn them well.

A point#

A point marks an exact posi­tion. It has no length, no width and no thick­ness. When you press the sharp tip of a pen­cil on paper, the tiny dot shows where a point is.

We name a point with a cap­i­tal let­ter, such as $A$, $B$ or $P$. We say "point $A$".

A line seg­ment#

Mark two points $A$ and $B$ on a sheet of paper. Join them with a ruler. The straight path from $A$ to $B$ is called a line seg­ment.

• $A$ and $B$ are its end points.
• It is the short­est path from $A$ to $B$.
• It has a def­i­nite length, which you can mea­sure with a ruler.

We write the line seg­ment $AB$ as $\overline{AB}$. The seg­ment $BA$ is the same seg­ment.

The edge of a ruler, the edge of a book and a stretched string between two nails are all mod­els of line seg­ments.

A line#

Now imag­ine the seg­ment $AB$ stretched on and on, beyond $A$ and beyond $B$, with­out ever end­ing. This is a line.

• A line has no end points.
• It goes on for ever in both direc­tions.
• So it has no def­i­nite length. We can­not mea­sure a line.

We write the line through $A$ and $B$ as $\overleftrightarrow{AB}$. The arrows at both ends show that it goes on both ways. Some­times a line is named with one small let­ter, such as line $l$ or line $m$.

A ray#

A ray starts at one point and goes on for ever in one direc­tion only.

• It has one end point, called its ini­tial point.
• It has no def­i­nite length.

The ray that starts at $A$ and passes through $B$ is writ­ten $\overrightarrow{AB}$. The ini­tial point is always writ­ten first. So $\overrightarrow{AB}$ and $\overrightarrow{BA}$ are dif­fer­ent rays: one starts at $A$, the other starts at $B$, and they point oppo­site ways.

A beam of light from a torch, or the rays of the sun, are mod­els of rays.

Com­par­ing the three#

Lines through points#

• Through one point you can draw as many lines as you like. Try it: draw a dot, then draw lines through it in dif­fer­ent direc­tions. There is no end to them.
• Through two dif­fer­ent points you can draw exactly one line.

Collinear points. Three or more points that lie on the same line are called collinear. Points that do not all lie on one line are called non-collinear.

Any two points are always collinear, because one line passes through both. So the ques­tion only mat­ters for three or more points.

Exam­ple 1#

Points $A$, $B$ and $C$ are such that $AB = 3$ cm, $BC = 5$ cm and $AC = 8$ cm. Are they collinear?

Notice that $AB + BC = 3 + 5 = 8$, which is exactly $AC$. The only way the path from $A$ to $B$ to $C$ can be as short as the straight path from $A$ to $C$ is if $B$ lies on the seg­ment $AC$. So the points are collinear, with $B$ between $A$ and $C$.

If $AB + BC$ had been more than $AC$, the path through $B$ would bend, and the points would not be collinear.

How lines meet#

Inter­sect­ing lines. Two lines that cross each other are called inter­sect­ing lines. The point where they cross is called the point of inter­sec­tion. Two dif­fer­ent lines can meet in at most one point.

Par­al­lel lines. Two lines on a flat sur­face that never meet, how­ever far they are drawn, are called par­al­lel lines. They stay the same dis­tance apart every­where. We write $l \parallel m$ and read it "$l$ is par­al­lel to $m$". The two rails of a rail­way track and the oppo­site edges of a ruler are mod­els of par­al­lel lines.

Con­cur­rent lines. Three or more lines that all pass through one point are called con­cur­rent lines. The spokes of a cycle wheel meet at its cen­tre like this.

A plane is a flat sur­face that goes on for ever in all direc­tions. The top of a table or a sheet of paper is a small part of a plane. All the lines in this chap­ter lie in one plane.

Count­ing line seg­ments#

Exam­ple 2#

Four points $A$, $B$, $C$ and $D$ lie on a line, in that order. How many line seg­ments do they make?

List them, start­ing from each point in turn and going to the points after it:

• From $A$: $\overline{AB}$, $\overline{AC}$, $\overline{AD}$, that is 3.
• From $B$: $\overline{BC}$, $\overline{BD}$, that is 2.
• From $C$: $\overline{CD}$, that is 1.

So there are $3 + 2 + 1 = 6$ line seg­ments.

With 5 points the count is $4 + 3 + 2 + 1 = 10$. In gen­eral, $n$ points make $\displaystyle \frac{n(n - 1)}{2}$ seg­ments. For 4 points that is $\displaystyle \frac{4 \times 3}{2} = 6$.

Prac­tice#

1. How many end points does a line seg­ment have? A ray? A line?
2. Which of these can be mea­sured with a ruler: a line, a ray or a line seg­ment?
3. How many lines can be drawn through one point? Through two dif­fer­ent points?
4. Is $\overrightarrow{PQ}$ the same as $\overrightarrow{QP}$? Give a rea­son.
5. Three points $P$, $Q$ and $R$ lie on a line, in that order. Name all the line seg­ments.
6. How many line seg­ments do 5 points on a line make?
7. $AB = 4$ cm, $BC = 6$ cm and $AC = 10$ cm. Are $A$, $B$ and $C$ collinear? Which point is in the mid­dle?
8. $PQ = 5$ cm, $QR = 4$ cm and $PR = 7$ cm. Are $P$, $Q$ and $R$ collinear?
9. Name one model of par­al­lel lines and one model of con­cur­rent lines that you can see at home or at school.

Answers#