Lines & Points

Plane Geometry is the study of figures which can be drawn using straight lines & curved lines on a plane.

Basic Ideas

The plane itself is assumed to extend in all directions without any boundary. In school, a sheet of paper laid on the top of a table is considered a plane.

There are 2 kinds of lines – Straight Line & Curved Lines. There are many kinds of curved lines. In modern times, curved lines are just called curves.

Until the advent of modern math, especially topology, the definition of a curve included anything which was not straight, thus including zigzag lines. Current understanding is that in a curve the change in the slope is a continuous one. (and this is in some way connected to the continuity of the real number line)

We restrict lines to those which are fully on the plane. I.e they are drawn on the plane using a pencil and a straight edge or a compass. The tip of the pencil represents a point and a straight edge represents a straight line.

Measure of a Line

Today we measure the length of a line in standard units like cm, m & km and express it in some numbers. Greek mathematicians who developed geometry did not use such measuring units in plane geometry.

Expressing them in numerical units was not needed for understanding geometrical concepts or for proving theorems. Plane geometry talks about the length of a line only in terms of magnitude or size.

These magnitudes are compared in 2 ways; additively & multiplicatively. Firstly they are expressed in terms of more/ less & equal. Secondly their relation is expressed as ratios.

So when talking about the 3 sides of a triangle, we can say that the sum of any 2 sides of a triangle is more than the third side. Or that the sides are in the ratio 1:2:3. The measurement of a side (say 2 inches or 5 cm) is irrelevant.

Only in schools do we use measurements to actually construct geometrical figures, say a triangle with sides 10, 4 & 7 cm.

Straight Edge & Compass

The straight edge symbolizes a straight line in real life. It can be used to draw a straight line joining any 2 points on a plane. It is not marked in inches or cm like a ruler. Hence a straight edge cannot be used to draw a line of a particular length, say 20 cm.

Plane Geometry studies only curves which can either be drawn with a compass or as paths traced by a point which moves under explicit directions. Such a path is called the Locus of a point. For example, the circle is the locus of (or path traced by) a point at the end of one of the arms of a compass which moves freely when the other arm is fixed at a point on a plane.

Euclid defined a general line (straight or curved) to be "breadthless length" with a straight line being a line "which lies evenly with the points on itself". His definition has not been found very helpful and is never used in any of the theorems.

Archimedes defined a straight line as the shortest distance between 2 points. This definition is useful and easy to understand.

Euclid defined a point as that which has no parts, implying that it has no dimensions. This definition is also not very helpful. We can think of a point as indicating a location or the intersection of 2 lines.

Plane, line & point are abstract ideas. The sheet of paper and the point & line that we draw on it using a pencil, however sharp it may be, are only ‘representations’ of the idea or a plane, point or a line.

Straight Line

An intuitive idea of a straight line can be formed by students by stretching a piece of thread tightly with both their hands. This can also lead them to the idea that a straight line indicates the shortest distance between 2 points, whatever their location. The orientation of a straight line does not matter. Students can also see that a straight edge or the edge of a ruler represents a straight line.

A Common Misconception

Such practical activities are necessary to avoid some misconceptions that students form because of the use of the term “straight” in daily life in slightly different perspective.

1. The word “straight” is normally used in terms of walking or looking straight ahead

2. While drawing a straight line on the board, teachers usually draw it as a line either parallel or perpendicular to the horizontal.

3. Hence students consider a slanting line as the opposite of a straight line.

It has to be understood by students that a straight line is contrasted with a curved line. In daily language we classify straight lines by their orientation – vertical, horizontal, inclined etc. But these definitions are not geometrically valid.

Use of broom sticks & paper folding

For understanding the basic ideas involving lines& shapes, it is a good practice to work with thin (rigid) plastic sticks or broom sticks. This is because in primary school children are not fluent in drawing lines on papers. They would waste a lot of time rubbing & redrawing & sharpening pencils. This would considerably reduce their focus and the time available for understanding concepts. Working with sticks is much faster, less messy and provides more time for thinking.

Paper folding is another practical way to learn a lot of concepts in geometry.