Names, notation and rules for lines

In Discover you met points, lines, rays and segments by looking at torch beams and railway tracks. Now we tighten everything up. What exactly counts as a line? How do you name one so that nobody can misunderstand you? How many lines can pass through two points, or three? How do you measure a segment accurately, and what goes wrong when you do not? And what precisely makes two lines parallel or perpendicular?

These are the questions a Class 6 or 7 textbook answers, and they are the foundations of every angle, triangle and construction you will ever meet. Getting them exactly right now saves a lot of confusion later.

Try to define a point. "A small dot"? But a dot has a size, and a point has none. "A position"? Then what is a position? Every definition uses other words, which need their own definitions, and so on forever.

Mathematicians solved this more than 2,000 years ago by not defining a few starting ideas. They simply describe them clearly and agree to use them. In school geometry the three starting ideas are the point, the line and the plane. Everything else, such as a segment, a ray, an angle or a triangle, is then defined using these three.

Here is a way to feel why a point has no size. Take a segment 1 cm long and cut it in half: 0.5 cm. Halve again: 0.25 cm. Keep halving. After 10 halvings the piece is 1 ÷ 1,024 cm, about one hundredth of a millimetre. After 20 halvings it is about one millionth of a centimetre, far thinner than a hair. You could go on forever and never get down to zero. A point is what is left at the very end of that endless shrinking: position, but no size.

Geometry is written in a very compact language. The rules are simple:

• Points get capital letters: A, B, P, Q.
• Lines can be named by two points on them (line AB) or by a small letter (line l, line m).
• Segments are named by their two end points (segment AB).
• Rays are named by their end point first, then any other point on the ray (ray AB).

Textbooks also use small symbols drawn over the letters. Because those symbols are hard to type, this site writes the words instead. Here is what each symbol looks like, described in words.

Mark a single point A on a page. Draw a straight line through it. Now draw another, at a different slant. And another. You can keep going forever: through one point there are endlessly many lines, one for every direction.

Now mark two points, A and B. Try to draw two different straight lines that both pass through A and B. You cannot. As soon as a straight line passes through A and B, its direction is fixed. Through two different points there is exactly one line.

A ray has one end point and goes on forever in one direction. To name it, always write the end point first. Ray AB starts at A and goes through B and beyond. It includes the point A itself.

If point C lies on ray AB beyond B, then ray AB and ray AC are the same ray: same start, same direction. Changing the second letter to another point on the ray does not change the ray. Changing the first letter always does.

A segment is the only one of the three with a length, so it is the one you can measure, compare, add and subtract.

If point B lies on segment AC, somewhere between A and C, then the two smaller segments fit together exactly: AB + BC = AC. This "betweenness" rule is behind many textbook problems. If B is not on segment AC, then AB + BC is more than AC, because the straight path is the shortest.

There are three ways to compare two segments, and they get more reliable as you go:

1. By observation, just looking. Quick, but easily fooled, especially when the segments point in different directions or have decorations at the ends.
2. By tracing, copying one segment onto thin paper and laying it on the other. Better, but clumsy and not very precise.
3. By measuring with a ruler, or with a divider and a ruler. This gives an actual number you can write down and check.

Rulers have thickness. The scale is printed on top of the plastic, but the segment is on the page underneath. If you look at the ruler from the side, the mark you see lined up with the end of the segment is not the right one: it shifts, and your reading is off by a millimetre or more. This is called parallax error.

The cure is simple: put your eye directly above the point you are reading. Even better, stand the ruler on its edge so that the markings touch the paper, or use a divider, which touches the paper with sharp points.

Three or more points that all lie on one straight line are called collinear ("co" means together, "linear" means on a line). If no single straight line passes through all of them, they are non-collinear.

Any two points are always collinear, because a line passes through any two points. The question only becomes interesting with three or more points.

How many lines can you draw through pairs of three points? It depends on whether they are collinear:

• Collinear A, B, C: lines AB, BC and AC are all the same line. Only 1 line.
• Non-collinear A, B, C: lines AB, BC and CA are all different. 3 lines, forming the sides of a triangle.

No other answer is possible: either all three points share one line, or no two of the three lines coincide.

Two lines in a plane that share a point are intersecting lines, and the shared point is their point of intersection. Why can they share only one point? Because if they shared two points, there would be two different lines through the same two points, and we know there is exactly one. So two distinct lines meet in at most one point.

When three or more lines all pass through the same point, they are called concurrent lines, and the shared point is the point of concurrence. Think of the spokes of a bicycle wheel meeting at the hub, or the folds you get when you fold a paper circle in half several times.

In Discover we said parallel lines "never meet". The precise definition adds one important condition: two lines are parallel if they lie in the same plane and never meet, however far they are extended.

Why insist on "the same plane"? Look at the edge where your classroom's front wall meets the ceiling, and the edge where a side wall meets the floor. They never meet. But they are not parallel either: one runs across the room, the other runs along it, at different heights. Lines like this, in different planes, are called skew lines. You will meet them properly in the Extend layer.

How do you measure the gap between two parallel lines? Not along any slanting segment you like: that would give different answers depending on the slant. The distance is measured along a segment that is perpendicular to both lines. That shortest gap is the same everywhere along parallel lines, which gives a practical test: measure the perpendicular gap at two places far apart. If the gaps are equal, the lines are parallel.

Two lines are perpendicular if they intersect at a right angle, 90°. In a diagram a right angle is marked with a small square drawn in the corner, instead of the usual arc. We write AB ⊥ CD.

When two lines are perpendicular, all four angles at the crossing are right angles. You only need to check one: if one angle is 90°, the one next to it along the straight line must be 180° − 90° = 90° too, and so on round.

A special perpendicular line is the perpendicular bisector of a segment. "Bisect" means cut into two equal parts. The perpendicular bisector of segment AB is the line that passes through the midpoint of AB and is perpendicular to it.

You can make one without any tools: draw segment AB on thin paper and fold the paper so that A lands exactly on B. Crease it. The crease passes through the midpoint (the two halves match) and meets AB at a right angle (the two angles at the fold match and add to 180°). Every point on that crease is the same distance from A as from B.

The precise language of this lesson is used every day by people who draw plans. An architect's floor plan, an electrician's wiring layout and a map all rely on the same few ideas: segments with exact lengths, lines that are parallel or perpendicular, and points named so that nobody can be confused.

A map grid is a set of evenly spaced parallel lines running north–south, crossed at right angles by another set running east–west. Every square has a name made from its column and row, like a seat in a cinema. A place is found by reading its column first and then its row. Official topographic sheets, such as those of the Survey of India used for planning roads and railways, are printed with a national grid like this.

Old Jaipur, planned in 1727 under Sawai Jai Singh II, was laid out with wide straight roads crossing at right angles into large blocks, so even the city itself is a grid of perpendicular lines.