Points, lines, line segments and rays

Before we can talk about shapes, we need to talk about the tiny ingredients that make shapes , points, lines, line segments, and rays. These are the alphabet of geometry.

Concept

A point is a precise location with no size at all. The tip of a sharp pencil, the dot of a sharpie, or the very tip of a needle gives an idea , but a true mathematical point has no thickness whatsoever. We mark a point with a small dot and label it with a capital letter such as $A$, $B$, $P$, or $T$.

A line is straight, has no thickness, and extends forever in both directions. We draw it with arrowheads on both ends to remind us that it never stops. A line through two points $A$ and $B$ is written $\overleftrightarrow{AB}$. Sometimes a single letter like $\ell$ or $m$ is used for a line. The most important fact: through any two distinct points there is exactly one line.

A line segment is the part of a line lying between two points. The two points are called its endpoints. The segment with endpoints $A$ and $B$ is written $\overline{AB}$. Unlike a line, a line segment has a finite length you can measure with a ruler.

A ray is a hybrid , it has one endpoint and then goes on forever in one direction, like the beam from a torch. A ray starting at $A$ and going through $B$ is written $\overrightarrow{AB}$. Note: $\overrightarrow{AB}$ and $\overrightarrow{BA}$ are different rays, because they start at different points and head in opposite directions.

How are these four objects related? Imagine a long, straight road. The road itself is a line. A section of the road from a temple to a bus stand is a line segment. A road that starts at a railway station and runs straight all the way north to the mountains, never ending, is a ray. A single milestone on the road is a point.

Two lines that never meet, no matter how far they are extended, are called parallel lines (think of railway tracks). Two lines that meet at a point are called intersecting lines. Two lines that meet at a right angle ($90°$) are called perpendicular lines.

Worked examples

Example 1. Three points $A$, $B$, $C$ are marked on a paper, all on the same line. How many line segments can you form using two of them?

• Pairs: $\overline{AB}$, $\overline{BC}$, $\overline{AC}$.
• So $3$ line segments.

Example 2. Are $\overrightarrow{PQ}$ and $\overrightarrow{QP}$ the same ray?

• No. $\overrightarrow{PQ}$ starts at $P$ and goes through $Q$ forever.
• $\overrightarrow{QP}$ starts at $Q$ and goes through $P$ forever.
• They start at different points and go in opposite directions.

Example 3. Through how many lines can you draw passing through a single given point $A$?

• Infinitely many. A line needs two points to be fixed; one is not enough.

Example 4. Two lines $\ell$ and $m$ intersect. In how many points can they intersect?

• Two distinct straight lines can intersect at most in one point. (If they shared two points, they would actually be the same line.)
• So the answer is exactly one point.

Try it yourself

1. Draw a line and mark three points $X$, $Y$, $Z$ on it. List all rays you can form with these points.
2. How many line segments can be formed using $5$ points, no three on the same line? (Hint: count pairs.)
3. Draw two parallel lines and a third line cutting across them. How many points of intersection?
4. Are the edges of a notebook parallel, perpendicular, or both?
5. Draw a ray $\overrightarrow{AB}$ and a different ray $\overrightarrow{AC}$ starting at the same point $A$. What kind of figure have you drawn?
6. Through two given points, how many lines pass?
7. Look around your classroom , write down two examples each of parallel and perpendicular lines.

Activity

Three-point investigation. Take a sheet of paper and mark $4$ points so that no three lie on the same straight line. With a ruler, draw every line segment that joins two of these points. How many line segments are there? Repeat with $5$ points and then $6$ points. Make a table and look for a pattern. (Hint: with $n$ points the answer is $\frac{n(n-1)}{2}$.)