The Building Blocks of Geometry

The Building Blocks of Geometry

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.

Terms

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

Points

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.

Problems

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.

Lines

Lines

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: Line AB

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.

Line Segments

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: Segment AB

Rays

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.

Problems Problem 2.1: Decide whether the following is a point, line, segment, ray, or none of these.

A line

Problem 2.2: Decide whether the following is a point, line, segment, ray, or none of these.

None of these


Segment


Ray


None of these

Planes

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.

Problems

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

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.

Problems Problem 4.1: How many dimensions does the following figure have?

One

Problem 4.2: How many dimensions does the following figure have?

Two


One


Zero


Two

Problem 4.6: If a region in space is spanned by two rays, how many dimensions does it have?

Two

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.

Space 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.

Problems 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?

Three

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.

The Building Blocks of Geometry

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