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August 13, 2008

Geometry Notes — Undefined Terms and Other Basic Terms

Filed under: Geometry — Tags: — bowman @ 10:33 am

The 3 “undefined terms in geometry” are the building blocks for the rest of the subject. They are like the foundation that is at the bottom of a house – without a strong foundation, the rest of the house could not be built.

The 3 terms are point, line, and plane.

Points are the basis of all geometry. Points are zero-dimensional. That basically means that they have no height, length, or width. They are just there.
A point is like a star in the night sky. It is a little spec in a large endless sky. However, unlike stars, geometric points have no size. Think of them as being so small that they take up zero amount of space.
A point shows location. A point is represented by a dot and is usually named with a letter of choice.
Points are named by capital letters. Remember, points have no size, but we draw them as a “dot” on our papers so that we know where they are. Any letter can be used to name a point.
A point shows an exact position or location on a plane surface. It is important to understand that a point is not a thing, but a place. We draw a point by placing a dot with a pencil. This dot may have a diameter of 0.5 mm, but a point has no size. No matter how far you zoomed in, it would still have no width.

A line is a one-dimensional figure. That is, a line has length, but no width or height. Basically, a line is made up of an infinite number of points. Points in the same line are called collinear.
A line is like the edge of a ruler that never ends. Take a ruler and draw a line – now imagine if that line kept going straight forever. The line you have is thick enough for you to see, but you need to imagine that your line is so thin that you can’t see it – it has no thickness at all. That is a geometric line.
We draw arrows at the end of the line. This tells us that our line extends forever without having to draw it to the edge of our paper every time.
A line is identified by naming two points on the line or by writing a lowercase letter of choice after the line. The notation, for example, AB (written with a line symbol <—> over the letters), is read as “line AB” and refers to the line that has points A and B.

Planes are two-dimensional. A plane has length and width, but no height, and extends infinitely on all sides. Planes are thought of as flat surfaces, like a table top. A plane is made up of an infinite amount of lines.
A plane is a flat piece of land (like a football field) that extends forever. Imagine that you can pick that football field up, and put it anywhere in the air that you like. You can even turn it side ways, or diagonally. A plane is just a flat edge (like a piece of paper) that has no thickness (just like the line) and extends forever.
It is a two-dimensional object.
Just like lines, planes too can be named in two different ways. The first way is to name it with a capital scripted (cursive) letter. Planes can also be named by naming any three non-collinear points that are within the plane.

Take out a sheet of paper. Draw a dot on that piece of paper – that will be your first point. Outstanding. Next, draw another dot to represent point number two. Once you have the two points drawn, take a ruler and draw a straight line through those points. Make sure your line goes to the edge of your piece of paper.
Pretend that your piece of paper goes on forever in all directions. Pretend that the line you have drawn never stops either. After you’ve done this, here’s what you have…
The dots are geometric points.
The line drawn through the points is a geometric line.
The piece of paper is a plane.

Space is the set of all points. It is made up of an infinite number of planes.

Collinear points are points that are all on the same line.

“between” is a very simple word that is important in geometry. For a point to be between two other points, all three points must be collinear.

Coplanar points are points that are all on the same plane.

Coplanar lines are lines that are all on the same plane.

Intersection – The set of points two figures have in common.

  • when 2 lines intersect, the intersection is a single point
  • when a line intersects a plane, the intersection is a single point
  • when 2 planes intersect, the intersection is a line

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