THE NATURE OF GEOMETRY Copyright © Cengage Learning. All rights reserved. 7
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THE NATURE OF GEOMETRY
Copyright © Cengage Learning. All rights reserved.
7
Copyright © Cengage Learning. All rights reserved.
7.1 Geometry
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Greek (Euclidean) Geometry
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Greek (Euclidean) Geometry
Geometry involves points and sets of points called lines, planes, and surfaces. Certain concepts in geometry are called undefined terms.
For example, what is a line? Is it a set of points? Any set of points? What is a point?
1. A point is something that has no length, width, or thickness.
2. A point is a location in space.
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Greek (Euclidean) Geometry
Certainly these are not satisfactory definitions because they involve other terms that are not defined. We will therefore take the terms point, line, and plane as undefined.
Geometry can be separated into two categories:
1. Traditional (which is the geometry of Euclid)
2. Transformational (which is more algebraic than the traditional approach)
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Greek (Euclidean) Geometry
When Euclid was formalizing traditional geometry, he based it on five postulates, which have come to be known as Euclid’s postulates.
A postulate or axiom is a statement accepted without proof. In mathematics, a result that is proved on the basis of some agreed-upon postulates is called a theorem.
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Greek (Euclidean) Geometry
The first four of these postulates were obvious and noncontroversial, but the fifth one was different.
This fifth postulate looked more like a theorem than a postulate.
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Greek (Euclidean) Geometry
It was much more difficult to understand than the other four postulates, and for more than 20 centuries mathematicians tried to derive it from the other postulates or to replace it by a more acceptable equivalent.
Two straight lines in the same plane are said to be parallel if they do not intersect.
Today we can either accept the fifth postulate as a postulate (without proof) or deny it. If it is denied, it turns out that no contradiction results; in fact, if it is not accepted, other geometries called non-Euclidean geometries result.
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Greek (Euclidean) Geometry
If it is accepted, then the geometry that results is consistent with our everyday experiences and is called Euclidean geometry.
Let’s look at each of Euclid’s postulates. The first one says that a straight line can be drawn from any point to any other point.
To connect two points, you need a device called a straightedge (a device that we assume has no markings on it; you will use a ruler, but not to measure, when you are treating it as a straightedge).
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Greek (Euclidean) Geometry
The portion of the line that connects points A and B in Figure 7.3 is called a line segment.
We write AB (or BA). We contrast this notation with , which is used to name the line passing through the points A and B. We use the symbol |AB| for the length of segment AB.
Portions of lines
Figure 7.3
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Greek (Euclidean) Geometry
The second postulate says that we can draw a straight line. This seems straightforward and obvious, but we should point out that we will indicate a line by putting arrows on each end.
If we consider a point on a line, that point separates the line into parts: two half-lines and the point itself. If the arrow points in only one direction, the figure is called a ray.
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Greek (Euclidean) Geometry
We write AB (or BA) for the ray with endpoint A passing through B. These definitions are illustrated in Figure 7.3.
To construct a line segment of length equal to the length ofa given line segment, we need a device called a compass.
Figure 7.3
Portions of lines
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Greek (Euclidean) Geometry
Figure 7.4 shows a compass, which is used to mark off andduplicate lengths, but not to measure them.
A compass
Figure 7.4
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Greek (Euclidean) Geometry
If objects have exactly the same size and shape, they are called congruent. We can use a straightedge and compass to construct a figure so that it meets certain requirements.
To construct a line segment congruent to a given line segment, copy a segment AB on any line .
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Greek (Euclidean) Geometry
First fix the compass so that the pointer is on point A and the pencil is on B, as shown in Figure 7.5a.
Constructing a line segment
Figure 7.5 (a)
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Greek (Euclidean) Geometry
Then, on line , choose a point C. Next, without changing the compass setting, place the pointer on C and strike an arc at D, as shown in Figure 7.5b.
Constructing a line segment
Figure 7.5(b)
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Greek (Euclidean) Geometry
Euclid’s third postulate leads us to a second construction. The task is to construct a circle, given its center and radius. These steps are summarized in Figure 7.6.
c. Hold the pointer at point O and move the pencil end to draw the circle.
Construction of a circle
Figure 7.6
b. Set the legs of the compass on the ends of radius AB; move the pointer to point O without changing the setting.
a. Given, a point and a radius of length |AB|.
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Greek (Euclidean) Geometry
The fourth postulate is used when we consider angles.
The final construction of this section will demonstrate the fifth postulate. The task is to construct a line through a point P parallel to a given line , as shown in Figure 7.7a.
Given line
Construction of a line parallel to a given line through a given point
Figure 7.7 (a)
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Greek (Euclidean) Geometry
First, draw any line through P that intersects , at a point A, as shown in Figure 7.7b.
Construction of a line parallel to a given line through a given point
Figure 7.7 (b)
Draw line
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Greek (Euclidean) Geometry
Now draw an arc with the pointer at A and radius AP, and label the point of intersection of the arc and the line X, as shown in Figure 7.7c.
Construction of a line parallel to a given line through a given point
Figure 7.7 (c)
Strike arc
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Greek (Euclidean) Geometry
With the same opening of the compass, draw an arc first with the pointer at P and then with the pointer at X. Their point of intersection will determine a point Y (Figure 7.7d).Draw the line through both P and Y. This line is parallel to .
Construction of a line parallel to a given line through a given point
Figure 7.7 (d)
Determine point Y
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Transformational Geometry
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Transformational Geometry
Transformational geometry deals with the study of transformations. A transformation is the passage from one geometric figure to another by means of reflections, translations, rotations, contractions, or dilations. For example, given a line L and a point P, as shown in Figure 7.8, we call the point P the reflection of P about the line L if PP is perpendicular to L and is also bisected by L.
A reflection
Figure 7.8
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Transformational Geometry
Each point in the plane has exactly one reflection point corresponding to a given line L. A reflection is called a reflection transformation, and the line of reflection is called the line of symmetry.
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Transformational Geometry
The easiest way to describe a line symmetry is to say that if you fold a paper along its line of symmetry, then the figure will fold onto itself to form a perfect match, as shown inFigure 7.9.
Line symmetry on the maple leaf of Canada
Figure 7.9
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Transformational Geometry
Other transformations include translations, rotations, dilations, and contractions, which are illustrated in Figure 7.11.
Transformations of a fixed geometric figure
Figure 7.11
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Similarity
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Similarity
Geometry is also concerned with the study of the relationships between geometric figures. A primary relationship is that of congruence. A second relationship is called similarity.
Two figures are said to be similar if they have the same shape, although not necessarily the same size.
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