Geometry for Elementary School/Points

Geometry for Elementary School
Concepts Points Lines

A point is a dot that is so small that its height and width are actually zero! This may seem too small. So small that no such thing could ever really exist. But it does fit with our intuition about the world. Even though everything in the physical world around us of things larger than atoms, it is still very useful to talk about the centers of these atoms, or electrons. A point can be thought of as the limit of dots whose size is decreasing.

You might be surprised to learn that these are not points:
The reason that this shape is not a point is that it is too large, it has area. This is a 'circle'.
Even when taking a circle of half that size we don't get a point.
And that is too large as well...

A point is so small that even if we divide the size of these dots by 100, 1,000 or 1,000,000 it would still be much larger than a point. A point is considered to be infinitely small. In order to get to the size of a point we should keep dividing the circle size by two – forever. Don't try it at home.

A point has no length, width, or depth. In fact, a point has no size at all. A point seems to be too small to be useful. Luckily, as we will see when discussing lines we have plenty of them. It may be best to think of a point as a location, as in a location where two lines cross.

Why define a point as an infinitely small dot? For one thing it has a very precise location, not just the center of a rough dot, but the point itself. Another reason is that if the drawing is made much bigger or smaller the point stays the same size. A point which is an infinitely small dot would be too small to see, so we must use a big old visible normal dot, or where two lines cross to represent it and its approximate location on paper.

When we name a point, we always use an uppercase letter. Often we will use for "point" if we can, and if have more than one dot, we will work our way through the alphabet and use , , and so on. However, nowadays many people will start with any letter they like, although the still remains the best way.

If some points are on the same line, we call them 'colinear'. If they are on the same plane, they are 'coplanar'. Two points are always colinear. But a point can be collinear with several points.Two to three points are always coplanar. Of course this is tautological since the definition of a 'line' is 'two connected points', and the definition of a 'plane' is 'the surface specified by three points'.



1 Which of the following does a point have?


2 A collection of points are colinear if they are:

On the same surface as each other.
On the same circle as each other.
On the same line as each other.

3 A collection of points are coplanar if they are:

On the same flat surface as each other.
On the surface of the same cube.
On the surface of the same cone as each other.