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Basic Geometric Definitions

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The Three Dimensions

It's not too hard to understand zero, one, two, and three dimensions. What about four dimensions? What is the fourth dimension?

Watch This

http://www.youtube.com/watch?v=VQ15ECqYDGs James Sousa: Points, Lines, Planes

Guidance

You live in a three dimensional world. Solid objects, such as yourself, are three dimensional. In order to better understand why your world is three dimensional, consider zero, one, and two dimensions:

A point has a dimension of zero. In math, a point is assumed to be a dot with no size (no length or width).

A line or line segment has a dimension of one. It has a length . A number line is an example of a line. To describe a point on a number line you only need to use one number. Remember that by definition, a line is straight.

A shape or a plane has a dimension of two. A shape like a rectangle has length and width . The rectangular coordinate system that you create graphs on is an example of a plane. To describe a point on the rectangular coordinate system you need two numbers, the $x$ -coordinate and the $y$ -coordinate.

A solid has a dimension of three. It has length, width, and height . You can turn the rectangular coordinate system into a three dimensional coordinate system by creating a third axis, the $z$ -axis, that is perpendicular to both the  $x$ and  $y$ axes. Because paper and screens have dimensions of two , it is hard to represent three dimensional objects on them. Artists use perspective techniques to allow the viewer to imagine the three dimensions.

You can think of the dimension of a space as the number of numbers it would take to describe the location of a point in the space. A single point on its own has dimension zero. A line, such as a number line, has dimension one. A plane, such as the rectangular coordinate system, has dimension two. A solid, such as a cube, has dimension three.

Example A

You graph a line on a rectangular coordinate system. How many dimensions does that line have?

Solution: Even though the rectangular coordinate system has two dimensions, the line itself has only one dimension.

Example B

How many points make up a line?

Solution: A line is made up of an infinite number of points.

Example C

Is the edge of a desk best described as a point, a line, a plane, or solid?

Solution: The edge of a desk is best described as a line. It has one dimension.

Concept Problem Revisited

Four dimensions don't exist in our world, so it is very hard to imagine an object with dimension four. It helps to think about how two dimensions become three dimensions. A shape with dimension two moves up and down to create a solid with dimension three. Similarly, you can imagine a solid (such as a cube) with dimension three moving within itself to create a tesseract, with dimension four.

Vocabulary

In math, the dimension of a space is thought of as the number of numbers needed to label a point within it. A single point on its own has dimension zero. A line, such as a number line, has dimension one. A plane, such as the rectangular coordinate system, has dimension two. A solid, such as a cube, has dimension three.

Three or more points are collinear if they lie on the same line.

Points or lines are coplanar if they lie on the same plane.

Guided Practice

1. Name three points with dimension zero from the figure above.
2. Name three line segments with dimension one from the figure above.
3. Name three planes with dimension two from the figure above.

1. Any points that make up this prism will work. These points are called vertices. For example, point $A$ , point $B$ , point $C$ .
2. Any line segments that make up this prism will work. These line segments are called edges. For example, $\overline{AB}$ , $\overline{BC}$ , $\overline{CD}$ .
3. Any “sides” that make up this prism will work. These “sides” are called faces. You can name planes by three points in the plane (as long as those points are not all on the same line). For example, $ABC$ , $BCG$ , $CGH$ .

Practice

1. In your own words, explain why a line has a dimension of one and a plane has a dimension of two.

2. Give a real-world example of something with a dimension of one.

3. Give a real-world example of something with a dimension of two.

4. Give a real-world example of something with a dimension of three.

Use the figure below for #5-#6.

5. Points are considered coplanar if they lie on the same plane. What's an example of a point that is coplanar with points  $H$ and $E$ ?

6. What's an example of a point that is coplanar with points  $D$ and $E$ ?

Use the figure below for #7-#12.

7. Name three points from the figure above.

8. Name three line segments from the figure above.

9. Name three planes from the figure above.

10. Name a point that is coplanar with points  $A$ and $B$ .

11. Name another point that is coplanar with points  $A$ and $B$ , but not also coplanar with your answer to #10 such that all four points are on the same plane.

12. Name a point that is coplanar with  $C$ and $E$ .

13. A plane has a dimension of ____.

14. A line segment has a dimension of ____.

15. A cube has a dimension of ____.