Geometry for Elementary School/Points

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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 know, but these are not points:
Disc Plain black.svg
The reason that this shape is not a point is that it is too large, it has area. This is a 'ball'.
Disc Plain black.svg
Even when taking a ball of half that size we don't get a point.
Disc Plain black.svg
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 ball 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 P for "point" if we can, and if have more than one dot, we will work our way through the alphabet and use Q, R, and so on. However, nowadays many people will start with any letter they like, although the P 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. 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'.