Straight paths: points, lines, rays and segments

Look up from wherever you are sitting. How many straight edges can you count in ten seconds? The edge of a door. The top of a table. The lines in your notebook. The cable running along the wall. The edge of your phone.

Now go outside in your mind. A railway track running off into the distance. The white crease line on a cricket pitch. A kite string pulled tight on Makar Sankranti. The beam of a torch at night. The border of a badminton court painted on the floor of a school hall.

All of these are made of the same few ideas: points, lines, rays and line segments. They are the alphabet of geometry. Every shape you will ever draw, from a triangle to a rangoli to the plan of a house, is built from them.

Put the sharp tip of your pencil on a page and press very lightly. You have made a tiny dot. That dot shows a place on the page.

A point is exactly that: an exact position. The dot you drew is only a picture of a point. A real mathematical point is so small that it has no length, no width and no thickness at all. You cannot measure it. You can only say where it is.

That sounds strange, but you already use points like this. When a friend says "meet me at the school gate", you do not worry about how big the meeting place is. You only care where it is.

Why capital letters? So that we can talk about points without pointing. If a page has three dots, saying "that one, no, the other one" is confusing. Saying "point B" is clear.

Mathematicians agreed on a simple habit: points get capital letters. You will see this in every textbook in India and around the world. You can use any capital letter you like, but each point on a page should have its own letter.

Mark two points on a page, A and B. Put your ruler against both of them and draw the straight path from A to B. What you have drawn is a line segment.

A line segment is the straight path between two points. It starts at one point and stops at the other. Those two points are called its end points. Because it stops at both ends, a line segment has a length you can measure: 5 cm, 12 cm, 3.5 cm.

Here is something surprising. Of all the paths you could take from A to B (wiggly ones, zigzag ones, curved ones), the straight one is the shortest. That is why people cut across a grassy corner of a park instead of walking along the path: their feet know that a straight line is the shortest way.

Switch on a torch in a dark room. The light starts at the bulb and shoots out in one direction. If nothing got in its way, where would it stop? It would not. It would go on and on, out of the window, past the Moon, into space.

That is the idea of a ray. A ray has a starting point, but it goes on forever in one direction. Sunbeams, laser pointers and the beam from a lighthouse are all good pictures of rays.

Stand in an open field and point in any direction: north, east, towards a tree, towards a bird. Each direction you point in gives a different ray starting at you. How many directions are there? Endlessly many. So endlessly many rays can start from the same point.

Think of the Sun drawn by a small child: a circle with rays sticking out in every direction. The child draws eight or twelve, but the real Sun sends light out in every direction at once.

Take the segment AB and imagine stretching it beyond both ends, past A and past B, and never stopping in either direction. What you get is a line.

A line goes on forever both ways. It has no end points at all. We draw a line with an arrowhead at each end to show that it keeps going. We can name it using any two points on it, as line AB, or with a small letter, as line l.

Not every path is straight. The outline of a bangle, the path of a thrown ball, the letter S and the edge of a leaf are curves. In everyday speech we sometimes say "draw a curved line", but in geometry the word line always means straight. A path that bends is called a curve.

A curve that starts and ends at the same point, like a bangle or the letter O, is called a closed curve. One that does not, like the letter C, is an open curve.

Now think about a surface instead of a path. The top of a table, a sheet of paper, a blackboard, the floor of your classroom: they are all flat.

Imagine the top of your table growing wider and longer in every direction, forever, and staying perfectly flat and paper-thin. That is a plane. Every drawing you make in your notebook lives on a plane. Points, lines, rays and segments all sit on planes.

Open a pair of scissors. The two blades cross at the screw. Look at a road map: roads cross at junctions. Write the letter X: its two strokes cross in the middle.

When two lines cross, we say they intersect. The place where they cross is a single point called the point of intersection. Two straight lines can cross in only one point: once they have crossed, they are heading away from each other and can never meet again.

Look at the lines printed in your notebook. However far along the page you go, they never get closer together and never get further apart. Now picture a railway track: two steel rails, always the same distance apart, running for hundreds of kilometres. If the rails ever came together, the train would fall off!

Lines like these are called parallel lines. They lie on the same flat surface, they stay the same distance apart all the way along, and so they never meet, however far you extend them.

Look at the corner of a book, a door frame, a window, or the plus sign +. In each one, two straight edges meet to make a perfect square corner. That square corner is called a right angle, and it measures 90 degrees (written 90°).

Two lines that cross or meet at a right angle are called perpendicular lines. They are a special kind of intersecting line: they do not just meet, they meet "square".

Look at the edge of a still pond or the surface of water in a glass. It lies perfectly flat and level. A line that is level like this is called horizontal (from the word horizon, the line where the sky seems to meet the land or the sea).

Now hang a small stone on a thread and hold the thread still. The thread points straight down towards the centre of the Earth. A line that points straight up and down like this is called vertical. A horizontal line and a vertical line always meet at a right angle, so they are perpendicular.

Now that you know the names, you will start seeing them everywhere. Games are a good place to look, because courts and pitches are painted with lines that follow strict rules.

Even the letters you write are built from lines. Capital letters in a plain font are a great place to practise spotting parallel and perpendicular strokes.

Walk onto any sports ground in a school in India and you are standing on a giant geometry diagram. The lines are painted so that referees can make fair decisions: is the player in or out? Did the raider cross the line? Every one of those decisions depends on knowing exactly where a straight line is.

Now look at a zebra crossing. Each white stripe is a long, thin rectangle. On a standard crossing the stripes are parallel to one another, and the long side of each one runs along the road, the way the traffic moves; they are laid side by side across the road. The path you walk along to cross the road runs perpendicular to the road's edge: straight across, the shortest way to the other side. That is not an accident. The perpendicular path is the shortest one, so you spend the least time in the road.

Even the walls of your home hide lines. Electricians usually run the wires inside a wall in straight vertical and horizontal paths: straight up or down from a switch board, and level along the wall near the ceiling. That way, anyone who later hammers a nail into the wall knows where the wires are likely to be: never directly above or below a switch.