Geometry is essentially the study of shapes. In the world around us, every object we see is a shape of some kind. Some are simple, like a triangle, square, or circle. Others seem to be combinations of these simple shapes. It is the goal of this text to guide you through the study of the simple shapes that compose nearly everything in the world, so that you can apply these concepts to more complicated problems of geometry. To begin understanding these shapes, we need to first learn their components.
The simplest unit of geometry is the point. A collection of points in a certain array makes a line, and collections of lines in certain arrays create shapes, which may exist in a single plane, or may exist in more than one plane in space. These terms will be more carefully defined in the lessons that follow. For now it is just important to introduce you to the language of geometry. You'll become familiar with the names and properties of these figures, which are the building blocks of geometry, and learn to recognize them in more complex situations.
The dimension of a geometric figure determines whether it has length, area, volume, or none of the three. Dimension is perhaps the most important property of space. These concepts form the foundation of almost all higher geometry, and are important to understand from the start.
Colinear - Lying in one line (usually refers to a set of points).
Congruent - Of exactly the same size and shape; in other words, of exactly the same dimensions.
Coplanar - Existing in one plane (usually refers to points or lines).
Dimension - The number of lines required to span a region in space.
Line - A collection of points arrayed in a straight formation without limit.
Noncolinear - Not lying in one line (usually refers to a set of points).
Noncoplanar - Not existing in one plane (usually refers to points or lines).
Perpendicular - At a 90 degree angle (something is perpendicular relative to something else).
Plane - A flat, boundless surface in space.
Point - A specific location in space.
Ray - A portion of a line with a fixed endpoint on one end that extends without bound in the other direction.
Segment - A portion of a line with
A point is a way to describe a specific location in space. It is drawn by placing a single dot at the location you want to specify, and denoted in text with a capital letter. Below is pictured the point B.
Figure 1.1: The point B
A point has no length, or width. In real life it is not tangible; points are useful for identifying specific locations, but are not objects in themselves. They only appear so when drawn on a page.
Problem 1.1: What does a point specify?
A specific location in space.
Problem 1.2: How many points are there in space?
An infinite number.
A line is the infinite set of all points arrayed in a straight formation. It is difficult to formally define, but easy to understand. A line has no thickness, it has only length, and can be named by any two points on that line. For example, a line can be called line AB, or symbolized:
Figure 2.1: The symbol for line AB
A line can also be given a single letter as a name, such as p, and be called line p.
To form a line, take any two points, A and B, and draw a straight line through them. The line AB looks like this on paper:
Figure 2.2: LineAB
A line extends in both directions without bound; this is why lines are usually depicted with arrows on each end. Its length is infinite, and between any two points on a line, there lie an infinite number of other points. Do you see why? You can choose two points on a line that seem to lie very close to each other, but if you "zoom in" on these points, you can always identify a point halfway between them. Then you can repeat the process with one of the original "close" points and the new halfway point to identify another point in between the two "close" points. This way you can find an infinite number of points between any two points on a line.
Figure 2.3: Finding an infinite number of points on a line
Points are called colinear if they lie in the same line. Likewise, points are called noncolinear if they lie in different lines. Since a line is determined by two points, any two points are always colinear. When a group of three points is considered, however, they may be noncolinear. Colinearity is a relative term. Points are only colinear or noncolinear when considered with respect to other points. The figure below has a set of noncolinear points on the left, and a set of colinear points on the right.
Figure 2.4: Points A, B, and C are noncolinear, whereas points D, E, and F are colinear
Note, as stated above in the rule that any two points are colinear, that a line can be drawn through any of the two points in the diagram. Though point A is noncolinear with respect to D, E, and F, it is colinear with D.
A line segment is the portion of a line that lies between two points on that line, points A and B. Whereas a line has infinite length, a line segment has a finite length. A line segment is denoted by segment AB, or the symbol
Figure 2.5: The symbol for segment AB
Line segments of the same length are called congruent. A dash or a set number of dashes is drawn through congruent segments to symbolize their congruence. Here is a figure of a segment:
Figure 2.6: SegmentAB
A ray is a cross between a line and a line segment. It extends without bound in one direction, but not the other. It is determined by two points, one being the starting point for the ray, and the other determining the direction of the ray. A ray can be symbolized in the following way:
Figure 2.7: The symbol for ray AB
Below is a figure of a ray:
Figure 2.8: Ray AB
Just like lines, segments and rays have no thickness, only length. They are intangible, and only used to specify a set of locations in space.
Problem 2.1: Decide whether the following is a point, line, segment, ray, or none of these.
Problem 2.2: Decide whether the following is a point, line, segment, ray, or none of these.
None of these
Problem 2.3: Decide whether the following is a point, line, segment, ray, or none of these.
Problem 2.4: Decide whether the following is a point, line, segment, ray, or none of these.
Problem 2.5: Decide whether the following is a point, line, segment, ray, or none of these.
None of these
A plane is a boundless surface in space. It has length, like a line; it also has width, but not thickness. A plane is denoted by writing "plane P", or just writing "P". On paper, a plane looks something like this:
Figure 3.1: Plane P
There are two ways to form a plane. First, a plane can be formed by three noncolinear points. Any number of colinear points form one line, but such a line can lie in an infinite number of distinct planes. See below how different planes can contain the same line.
Figure 3.2: Many different planes can contain the same line
It takes a third, noncolinear point to form a specific plane. This point fixes the plane in position. The situation is something like a door being shut. Before the door is shut, it swings on hinges, which form a line. The door (a plane) can be opened to an infinite number of different positions, maybe just cracked a few inches, or maybe wide open (figures a, b in the diagram below). When the door is shut however, the wall on the other side of the hinges acts as the noncolinear third point and holds the door in place. At this point, the door represents one distinct plane (figure c).
Figure 3.3: A door is like a plane
The second way to form a plane is with a line and a point in that line. There are just two conditions. 1) the line must be perpendicular to the plane being formed (for an explanation of this concept, see Geometric Surfaces, Lines and Planes); 2) the point in the line must also be in the plane being formed. Given a line, a point in that line, and these conditions, a plane is determined.
Figure 3.4: A plane can be determined by a line and a point in that line, given certain conditions.
When points lie in the same plane, they are called coplanar. When points lie in different planes, they are called noncoplanar. The concept is much like that of colinearity.
As previously mentioned, a plane has no thickness. Remember that though the diagrams shown here make it appear otherwise, a plane also has no limits: it is an endless surface in space. Most of the geometry you will see in this guide will deal with plane geometry. We will deal with "flat" shapes that lie in a plane, and therefore have no thickness. All of the points in such geometric figures are coplanar.
Problem 3.1: What are the two ways to form a plane?
The first way is with three noncolinear points. All three lie in the plane that is formed. The second way is with a line and a point in that line. The line must be perpendicular to the desired plane, and the point in the line must be in the desired plane as well.
Problem 3.2: Is it possible to form a plane with just two points?
Yes, but only one of the points will lie in the plane. The two points must be used to form a line. This line must be set perpendicular to the desired plane, and the plane must contain one of the two points given.
Problem 3.3: What is the difference between coplanar and noncoplanar points?
Coplanar means that the points in question all lie in one plane. Noncoplanar means that the points in question do not all lie in one plane.
Dimension is a characteristic of all geometric regions, objects, and spaces. The previous sections have probably already made you aware of the concept of dimension. It is roughly the number of directions in which a region or object can be measured. More formally, it is the number of lines required to span a region in space. Examples make dimension much easier to understand.
A point is zero-dimensional. It has no length, width, thickness, or any other physical means of measurement. It only exists as a symbol to identify a single location in space.
A line is one-dimensional. It has the dimension of length. To put it another way, there is only one way that you can move along a line: lengthwise. In a similar vein, there is no way to move within a point. A point is a single location in itself, whereas a line is a collection of points, or locations.
A plane is two-dimensional. It has length and width. (Technically speaking, the property of width is really only length in a different direction). You can move along a plane in two directions, lengthwise and widthwise. You might think that you can actually move along a plane in an infinite number of directions, but actually every direction in which you move can be broken down into a component of length and a component of width.
Figure 4.1: Any ray in a plane can be divided into a component of length and width.
It should now be easier to understand the more formal definition of dimension: the number of lines required to span a region in space:
A point is not a region in space, it is only a specific location. Therefore it takes zero lines to span it, and it is zero-dimensional. One line is required to span a line (itself). Therefore a line is one- dimensional.It requires two lines to span a plane, so therefore a plane is two- dimensional. These two lines represent length and width. Any point in the plane can be expressed as a combination of a certain length and a certain width, depending on the location of the point. The span of a line (or many lines) is the region that contains all the points that can be expressed as combinations of that line (or lines). A geometric space can also be spanned by points or planes.
Problem 4.1: How many dimensions does the following figure have?
Problem 4.2: How many dimensions does the following figure have?
Problem 4.3: How many dimensions does the following figure have?
Problem 4.4: How many dimensions does the following figure have?
Problem 4.5: How many dimensions does the following figure have?
Problem 4.6: If a region in space is spanned by two rays, how many dimensions does it have?
Problem 4.7: If a region in space is spanned by an infinite number of points, how many dimensions does it have?
It could have any dimension. The dimension depends on the arrangement of the points. If they are all in a line, then the region is one-dimensional. If the points are noncoplanar, then the region is three-dimensional, etc.
So far we have dealt with points, lines, and planes. All of these lie within space. Space is the collection of all points. It has no shape, and it has no limits. Space is three-dimensional; that is, it has length, width, and a new dimension, height. Again, height is only length in a different direction. Height is a measurement of length perpendicular to length and width. Imagine a box: It has length, width, and height. It encloses space.
Figure 5.1: A box
A region in space can be either zero, one, two, or three-dimensional. A zero-dimensional region in space is a point. Instead of calling a point a region, because it is spanned by zero lines, it is more understandable to call it a location. Lines and planes are also regions in space. A three-dimensional object, like a ball of clay or a raindrop, also occupies a region in space. So does your shoe, your house, and your finger. Each is a collection of points.
These four building blocks of geometry, points, lines, planes, and space, form the basis for all of the geometry you will study in this guide. It is important to understand their properties fully. Depictions of points, lines, and planes on paper (or computer screens) often are misleading because they appear to add dimensions to the basic building blocks. Points appear to have dimension, lines appear to have width, and planes appear to have thickness. It is critically important to remember now and forever the true nature of these basic elements of geometry, so that they don't mislead you in the future when things are much more difficult to visualize.
Problem 5.1: Can two-dimensional objects exist in three-dimensional space?
Yes. They simply exist in the same plane within space.
Problem 5.2: What is the minimum number of lines it takes to span space?
Problem 5.3: Explain why three noncolinear points cannot form a three-dimensional object in space?
A plane can be created to contain any three points in space, therefore, three points isn't enough to create a three-dimensional object. It requires four noncoplanar points to form a three-dimensional object.