What are a point, a line, a line segment and a ray?

The building blocks of geometry

Every shape you will study in geometry, from triangles to circles, is built from a few basic ideas: points, line segments, lines and rays. Together with angles, they form the foundation of plane geometry.

Point

Mark a dot on paper with a sharp pencil. The sharper the tip, the thinner the dot. This tiny dot gives the idea of a point: a precise location with no length, breadth or height. The tip of a compass, the sharpened end of a pencil and the pointed end of a needle are models of a point.

When there are several points, we name each with a single capital letter, such as Z, P and T, and read them as ‘point Z’, ‘point P’ and ‘point T’. Remember that the dots stand for exact locations and must be imagined as invisibly thin.

Line segment

Fold a piece of paper and unfold it: the crease gives the idea of a line segment. Now mark two points A and B and try joining them by different routes, straight, curved or zigzag. The shortest route from A to B, including A and B, is the line segment AB. It is written AB or BA, with a bar over the letters, and A and B are its end points. A line segment has a definite length that you can measure with a ruler.

Line

Imagine the line segment AB extended beyond A in one direction and beyond B in the other, without any end. This is a line. You can never draw a complete picture of a line, because it goes on forever; we draw arrows at both ends to show this. A line through A and B is written AB with a double arrow over the letters, or named by a single small letter like l or m.

Two important facts:

• Through two points, exactly one line can be drawn.
• Through one point, infinitely many lines can be drawn, in every direction.

Ray

A ray is a portion of a line that starts at one point, its starting point or initial point, and goes on endlessly in one direction. A beam of light from a lighthouse, the light from a torch and the rays of the Sun are models of a ray.

To name a ray, write its starting point first and then any other point on it, with an arrow over the letters: ray AP starts at A and passes through P. The order matters:

• If ray OA also passes through B, it can be called OB as well: same start, same direction.
• But AO is a different ray: it starts at A and points towards O.

The four compared

	End points	Goes on forever?	Name
Point	(is a location)	No	P
Line segment	Two	No	AB or BA
Ray	One (starting point)	In one direction	AP (start first)
Line	None	In both directions	AB or l

Reading figures

In the book’s Fig. 2.4, five points are joined in order: L to M, M to P, P to Q and Q to R. The line segments are LM, MP, PQ and QR. The end points L and R lie on just one segment each; M, P and Q each lie on two.

In Fig. 2.5, three points are marked: A on one ray from T, and N and B on another. The rays are TA, TN and TB, and NB is a ray too. T is the starting point of TA, TN and TB, but not of NB, which starts at N. Notice that TN and TB point the same way along the same path.

Try it yourself

1. Mark one point and draw as many lines through it as you can. You will never run out.
2. Mark two points and try to draw two different straight lines through both. You cannot: only one fits.
3. Draw a line l, mark points E and F on it and a point D off it. Now say which points lie on l.

These ideas lead straight to angles, which are made of two rays: see what is an angle. For the whole chapter, see the Chapter 2 notes.