Download 9.2 Curves, Polygons, and Circles

502
CHAPTER 9
9.2
Geometry
Curves, Polygons, and Circles
Curves The basic undefined term curve is used for describing figures in the
plane. (See the examples in Figure 14.)
Simple; closed
Simple; not closed
Not simple; closed
Not simple; not closed
FIGURE 14
Simple Curve; Closed Curve
A simple curve can be drawn without lifting the pencil from the paper,
and without passing through any point twice.
A closed curve has its starting and ending points the same, and is also
drawn without lifting the pencil from the paper.
B
A
Convex
(a)
B
A
Not convex
A figure is said to be convex
if, for any two points A and B inside the figure, the
•–•
line segment AB (that is, AB ) is always completely inside the figure. Figure 15(a)
shows a convex figure while (b) shows one that is not convex.
Among the most common types of curves in mathematics are those that are both
simple and closed, and perhaps the most important of these are polygons. A polygon
is a simple closed curve made up only of straight line segments. The line segments are
called the sides, and the points at which the sides meet are called vertices (singular:
vertex). Polygons are classified according to the number of line segments used as
sides. The chart in the margin gives the special names. In general, if a polygon has
n sides, and no particular value of n is specified, it is called an n-gon.
Some examples of polygons are shown in Figure 16. A polygon may or may not
be convex. Polygons with all sides equal and all angles equal are regular polygons.
(b)
FIGURE 15
Convex
Classification of
Polygons According
to Number of Sides
Number
of Sides
3
4
5
6
7
8
9
10
Not convex
Polygons are simple closed curves made up of straight line segments.
Name
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
Regular polygons have equal sides and equal angles.
FIGURE 16
Triangles and Quadrilaterals
Two of the most common types of polygons
are triangles and quadrilaterals. Triangles are classified by measures of angles as
well as by number of equal sides, as shown in the following box. (Notice that tick
marks are used in the bottom three figures to show how side lengths are related.)
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9.2
Curves, Polygons, and Circles
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Types of Triangles
Tangrams The puzzle-game
above comes from China, where it
has been a popular amusement for
centuries. The figure on the left is
a tangram. Any tangram is
composed of the same set of
seven tans (the pieces making up
the square are shown on the right).
Mathematicians have
described various properties of
tangrams. While each tan is
convex, only 13 convex tangrams
are possible. All others, like the
figure on the left, are not convex.
All Angles Acute
One Right Angle
Acute triangle
Right triangle
All Sides Equal
Two Sides Equal
Equilateral triangle
Isosceles triangle
One Obtuse Angle
Angles
Obtuse triangle
No Sides Equal
Sides
Scalene triangle
Quadrilaterals are classified by sides and angles. It can be seen below that an
important distinction involving quadrilaterals is whether one or more pairs of sides
are parallel.
Types of Quadrilaterals
Sample Figure
A trapezoid is a quadrilateral with one pair of
parallel sides.
Trapezoid
A parallelogram is a quadrilateral with two pairs
of parallel sides.
Parallelogram
A rectangle is a parallelogram with a right angle
(and consequently, four right angles).
Rectangle
A square is a rectangle with all sides having
equal length.
Square
A rhombus is a parallelogram with all sides
having equal length.
Rhombus
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CHAPTER 9
Geometry
An important property of triangles that was first proved by the Greek geometers
deals with the sum of the measures of the angles of any triangle.
Angle Sum of a Triangle
The sum of the measures of the angles of any triangle is 180.
While it is not an actual proof, a rather convincing argument for the truth of this statement can be given using any size triangle cut from a piece of paper. Tear each corner
from the triangle, as suggested in Figure 17(a). You should be able to rearrange the
pieces so that the three angles form a straight angle, as shown in Figure 17(b).
(a)
(b)
FIGURE 17
(x + 20)°
(210 – 3x)°
x°
FIGURE 18
x x 20 210 3x 180
x 230 180
x 50
x 50
Combine like terms.
Subtract 230.
Divide by 1.
One angle measures 50, another measures x 20 50 20 70, and the third
measures 210 3x 210 350 60. Since 50 70 60 180, the
answers satisfy the angle sum relationship.
5
2
1
EXAMPLE 1
Find the measure of each angle in the triangle of Figure 18.
By the angle sum relationship, the three angle measures must add up to 180.
Write the equation indicating this, and then solve.
3
6
4
In the triangle shown in Figure 19, angles 1, 2, and 3 are called interior angles,
while angles 4, 5, and 6 are called exterior angles of the triangle. Using the fact that
the sum of the angle measures of any triangle is 180, and a straight angle also measures 180, the following property may be deduced.
FIGURE 19
Exterior Angle Measure
The measure of an exterior angle of a triangle is equal to the sum of the
measures of the two opposite interior angles.
In Figure 19, the measure of angle 6 is equal to the sum of the measures of angles 1
and 2. Two other such statements can be made. What are they?
EXAMPLE 2
Find the measures of interior angles A, B, and C of the triangle
in Figure 20 on the next page, and the measure of exterior angle BCD.
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Curves, Polygons, and Circles
505
B
(x + 20)°
A
(3x – 40)°
x°
C
D
FIGURE 20
By the property concerning exterior angles, the sum of the measures of interior
angles A and B must equal the measure of angle BCD. Thus,
x x 20 3x 40
2x 20 3x 40
x 60
x 60 .
Subtract 3x; subtract 20.
Divide by 1.
Since the value of x is 60,
Interior angle A 60
Interior angle B 60 20 80
Interior angle C 180 60 80 40
Exterior angle BCD 360 40 140.
Circles
One of the most important plane curves is the circle. It is a simple closed
curve defined as follows.
Circle
A circle is a set of points in a plane, each of which is the same distance
from a fixed point.
P
Q
O
T
R
FIGURE 21
A circle may be physically constructed with compasses, where the spike leg remains
fixed and the other leg swings around to construct the circle. A string may also be
used to draw a circle. For example, loop a piece of chalk on one end of a piece of
string. Hold the other end in a fixed position on a chalkboard, and pull the string taut.
Then swing the chalk end around to draw a circle.
A circle, along with several lines and segments, is shown in Figure 21. The
points P, Q, and R lie on the circle itself. Each lies the same distance from point O,
which is called
the center
of the circle. (It is the “fixed point” referred to in the defi•–• •–•
•–•
nition.) OP, OQ, and OR are segments whose endpoints are the center and a point
•–•
on the circle. Each is called a radius of the circle (plural: radii). PQ is
a segment
•–•
whose endpoints both lie on the circle and is an example of a chord. PR is a chord
that passes through the center and is called a diameter of the circle. –Notice that the
••
measure of a diameter is twice that of a radius. A diameter such as PR in Figure 21
divides a circle into two parts of equal size, each of which is called a semicircle.
i
RT is a line that touches (intersects) the circle in only one point, R, and is
i
called a tangent to the circle. R is the point of tangency. PQ, which intersects the
circle in two points, is called a secant line. (What is the distinction between a chord
and a secant?)
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CHAPTER 9
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Geometry
The portion of the circle shown in red in Figure 21 is an example of an arc of
the circle. It consists of two endpoints (P and Q) and all points on the circle “between” these endpoints. The colored portion is called arc PQ (or QP), denoted in
symbols as PQ (or QP).
The Greeks were the first to insist that all propositions, or theorems, about
geometry be given a rigorous proof before being accepted. According to tradition,
the first theorem to receive such a proof was the following.
Inscribed Angle
Any angle inscribed in a semicircle must be a right angle.
Thales made his fortune merely
to prove how easy it is to become
wealthy; he cornered all the oil
presses during a year of an
exceptionally large olive crop.
Legend records that Thales studied
for a time in Egypt and then
introduced geometry to Greece,
where he attempted to apply the
principles of Greek logic to his
newly learned subject.
To be inscribed in a semicircle, the vertex of the angle must be on the circle
with the sides of the angle going through the endpoints of the diameter at the base
of the semicircle. (See Figure 22.) This first proof was said to have been given by
the Greek philosopher Thales.
90°
90°
FIGURE 22
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Decide whether each statement is true or false.
5. A rhombus is an example of a regular polygon.
6. If a triangle is isosceles, then it is not scalene.
7. A triangle can have more than one obtuse angle.
8. A square is both a rectangle and a parallelogram.
9. A square must be a rhombus.
11. In your own words, explain the distinction between a
square and a rhombus.
10. A rhombus must be a square.
12. What common traffic sign in the U.S. is in the shape
of an octagon?
Identify each curve as simple, closed, both, or neither.
13.
14.
15.
16.