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JURUSAN MATEMATIKA
FAKULTAS MATEMATIKA DAN ILMU
PENGETAHUAN ALAM
UNIVERSITAS NEGERI SEMARANG
2012
Geometry
The term geometry is
derived from the Greek word geometria, meaning "to measure the Earth."
In its most basic sense, then, geometry was a branch of mathematics originally
developed and used to measure common features of Earth. Most people today know
what those features are: lines, circles, angles, triangles, squares,
trapezoids, spheres, cones, cylinders, and the like.
Humans have probably used
concepts from geometry as long as civilization has existed. But the subject did
not become a real science until about the sixth century b.c. At that point,
Greek philosophers began to express the principles of geometry in formal terms.
The one person whose name is most closely associated with the development of
geometry is Euclid
(c. 325–270 b.c.), who wrote a book called Elements. This work was the
standard textbook in the field for more than 2,000 years, and the basic ideas
of geometry are still referred to as Euclidean geometry.
Elements
of geometry
Statements.
Statements in geometry take one of two forms: axioms and propositions. An axiom
is a statement that mathematicians accept as being true without demanding
proof. An axiom is also called a postulate. Actually, mathematicians prefer not
to accept any statement without proof. But one has to start somewhere, and
Euclid began by listing certain statements as axioms because they seemed so
obvious to him that he couldn't see how anyone would disagree.
One axiom is that a single straight
line, and only one, can be drawn through two points. Another axiom is that two
parallel lines (lines running next to each other like train tracks) will never
meet, no matter how far they are extended into space. Indeed, mathematicians
accepted these statements as true without trying to prove them for 2,000 years.
Statements such as these form the basis of Euclidean geometry.
However, the vast majority of
statements in geometry are not axioms but propositions. A proposition is a
statement that can be proved or disproved. In fact, it is not too much of a
stretch to say that geometry is a branch of mathematics committed to proving
propositions.
Proofs. A
proof in geometry requires a series of steps. That series may consist of only
one step, or it may contain hundreds or thousands of steps. In every case, the
proof begins with an axiom or with some proposition that has already been
proved. The mathematician then proceeds from the known fact by a series of
logical steps to show that the given proposition is true (or not true).
Constructions. A
fundamental part of geometric proofs involves constructions. A construction in
geometry is a drawing that can be made with the simplest of tools. Euclid
permitted the use of a straight edge and a compass only. An example
of a straight edge would be a meter stick that contained no markings on it. A
compass is permitted in order to determine the size of angles used in a
construction.
Many propositions in geometry can be
proved by making certain kinds of constructions. For example, Euclid's first
proposition was to show that, given a line segment AB, one can construct an
equilateral triangle ABC. (An equilateral triangle is one with three equal
angles.)
Plane
A plane is a geometric
figure with only two dimensions: width and length. It has no thickness. The
flatness of a plane can be expressed mathematically by thinking about a
straight line drawn on the plane's surface. Such a line will lie entirely within
the plane with none of its points outside of the plane.
A plane extends forever in both
directions. Planes encountered in everyday life (such as a flat piece of paper
with certain definite dimensions) and in mathematics often have a specific
size. But such planes are only certain segments of the infinite plane itself.
Plane
and solid geometry
Euclidean geometry dealt
originally with two general kinds of figures: those that can be represented in
two dimensions (plane geometry) and those that can be represented in three
dimensions (solid geometry). The simplest geometric figure of all is the point.
A point is a figure with no dimensions at all. The points we draw on a piece of
paper while studying geometry do have a dimension, of course, but that
condition is due to the fact that the point must be made with a pencil, whose
tip has real dimensions. From a mathematical standpoint, however, the point has
no measurable size.
Perhaps the next simplest geometric
figure is a line. A line is a series of points. It has dimensions in one
direction (length) but in no other. A line can also be defined as the shortest
distance between two points. Lines are used to construct all other figures in
plane geometry, including angles, triangles, squares, trapezoids, circles, and
so on. Since a line has no beginning or end, most of the "lines" one
deals with in geometry are actually line segments—portions of a line that do
have a limited length.
In general, lines can have one of
three relationships to each other. They can be parallel, perpendicular, or at
an angle to each other. According to Euclidean geometry, two lines are parallel
to each other if they never meet, no matter how far they are extended.
Perpendicular lines are lines that form an angle of 90 degrees (a right angle,
as in a square or aT) to each other. And two lines that cross each other at any
angle other than 90 degrees are simply said to form an angle with each other.
Closed
figures. Lines also form closed figures, such as circles,
triangles, and quadrilaterals. A circle is a closed figure in which every part
of the figure is equidistant (at an equal distance) from some given point
called the center of the circle. A triangle is a closed figure consisting of
three lines. Triangles are classified according to the sizes of the angles
formed by the three lines. A quadrilateral is a figure with four sides. Some common
quadrilaterals are the square (in which all four sides are equal), the
trapezoid (which has two parallel sides), the parallelogram (which has two
pairs of parallel sides), the rhombus (a parallelogram with four equal sides),
and the rectangle (a parallelogram with four right- or 90-degree angles).
Solid
figures. The basic figures in solid geometry can be visualized
as plane figures being rotated through space. Imagine that a circle is caused
to rotate around its center. The figure produced is a sphere. Or imagine that a
right triangle is rotated around its right angle. The figure produced is a
cone.
Area
and volume
The fundamental principles
of geometry involve statements about the properties of points, lines, and other
figures. But one can go beyond those fundamental principles to express certain
measurements about such figures. The most common measurements are the length of
a line, the area of a plane figure, or the volume of a solid figure. In the
real world, length can be determined using a meter stick or yard stick.
However, the field of analytic geometry provides a way to determine the length
of a line by using principles adapted from geometry.
Mathematical formulas are available for determining
the area of any figures in geometry, such as rectangles, squares, various kinds
of triangles, and circles. For example, the area of a rectangle is given by the
formula A = l · h, where l is the length of the rectangle and h is its height.
One can find the areas of portions of solid figures as well. For example, the
base of a cone is a circle. The area of the base, then, is A = π · r2,
where π is a constant whose value is approximately 3.1416 and r is the radius
of the base. (Pi [π] is the ratio of the circumference of a circle to its
diameter, and it is always the same, no matter the size of the circle. The
circumference of a circle is its total length around; its diameter is the
length of a line segment that passes through the center of the circle from one
side to the other. A radius is a line from the center to any point on the
circle.)
Words to
Know
1.
Axiom: A mathematical statement accepted
as true without being proved.
2.
Construction: A geometric drawing that can be
made with simple tools, such as a straight edge and a compass.
3.
Euclidean geometry: A type of geometry based on certain
axioms originally stated by Greek mathematician Euclid.
4.
Line: A collection of points with one
dimension only—that of length.
5.
Line segment: A portion of a line.
6.
Non-Euclidean geometry: A type of
geometry based on axioms other than those first proposed by Euclid.
7.
Plane geometry: The study of geometric figures that
can be represented in two dimensions only.
8.
Point: A figure with no dimensions.
9.
Proposition: A mathematical statement that can
be proved or disproved.
10. Proof: A
mathematical statement that has been demonstrated logically to be correct.
11. Solid
geometry: The study of geometric figures that can be represented in three
dimensions.
Formulas for the volume of geometric
figures also are available. For example, the volume of a cube (a
three-dimensional square) is given by the formula V = s3, where s is
equal to the length of one side of the cube.
Other
geometries
With the growth of the
modern science of mathematics, scholars began to ask whether Euclid's initial
axioms were necessarily true. That is, would it be possible to imagine a world
in which more than one straight line could be drawn through two points. Such
ideas often sound bizarre at first. For example, can you imagine two parallel
lines that do eventually meet at some point far in the distance? If so, what
does the term parallel really mean?
Yet, such ideas have
turned out to be very productive for the study of certain special kinds of
spaces. They have been given the name non-Euclidean geometries and are used to
study certain kinds of mathematical, scientific, and technical problems.
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