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Copyright © 2005 Pearson Education, Inc.
SEVENTH EDITION and EXPANDED SEVENTH EDITION
Slide 9-1
Chapter 9
Geometry
Copyright © 2005 Pearson Education, Inc.
9.1
Points, Lines, Planes, and
Angles
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Basic Terms
„
A line segment is part of a line between two
points, including the endpoints.
Description
Diagram
Line AB
Ray AB
A
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B
B
A
Ray BA
Line segment AB
Symbol
B
A
A
B
suur
AB
uuur
AB
uuur
BA
AB
Slide 9-4
Plane
„
„
„
„
Any three points that are not on the same line
(noncollinear points) determine a unique plane.
A line in a plane divides the plane into three
parts, the line and two half planes.
Any line and a point not on the line determine a
unique plane.
The intersection of two planes is a line.
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Slide 9-5
Angles
„
„
„
The measure of an angle is the amount of rotation from
its initial to its terminal side.
Angles can be measured in degrees, radians, or,
gradients.
Angles are classified by their degree measurement.
‰
‰
‰
‰
Right Angle is 90°
Acute Angle is less than 90°
Obtuse Angles is greater than 90° but less than 180 °
Straight Angle is 180°
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Slide 9-6
Types of Angles
„
„
„
Adjacent Angles-angles that have a common
vertex and a common side but no common
interior points.
Complementary Angles-two angles whose sum
is 90 degrees.
Supplementary Angles-two angles whose sum
is 180 degrees.
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Slide 9-7
Example
„
If ABC and ABD are supplementary and the
measure of ABC is 6 times larger than CBD,
determine the measure of each angle.
C
m ABC + m CBD = 180o
6 x + x = 180
o
7 x = 180
o
x = 25.7o
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A
B
D
m ABC = 154.2o
m ABD = 25.7o
Slide 9-8
More definitions
„
„
„
Vertical angles have the same measure.
A line that intersects two different lines, at two
different points is called a transversal.
Special angles are given to the angles formed
by a transversal crossing two parallel lines.
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Slide 9-9
Special Names
Alternate interior
angles
Alternate exterior
angles
Corresponding
angles
Interior angles on the
opposite side of the
transversal—have the
same measure
Exterior angles on the
opposite sides of the
transversal—have the
same measure
One interior and one
exterior angles on the
same side of the
transversal-have the same
measure
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1 2
3 4
5 6
7 8
1 2
3 4
5 6
7 8
1 2
3 4
5 6
7 8
Slide 9-10
9.2
Polygons
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Polygons
„
Polygons are names according to their number of sides.
Number of
Sides
Name
Number of
Sides
Name
3
Triangle
8
Octagon
4
Quadrilateral
9
Nonagon
5
Pentagon
10
Decagon
6
Hexagon
12
Dodecagon
7
Heptagon
20
Icosagon
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Slide 9-12
Triangles
„
The sum of the measures of the interior angles
of an n-sided polygon is
(n − 2)180°.
„
Example: A certain brick paver is in the shape
of a regular octagon. Determine the measure of
an interior angle and the measure of one
exterior angle.
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Slide 9-13
Triangles continued
„
Determine the sum of the
interior angles.
„
S = (n − 2)180o
= (8 − 2)(180o )
„
= 6(180o )
= 1080o
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„
The measure of one
interior angle is
1080o
o
= 135
8
The exterior angle is
supplementary to the
interior angle, so
180° − 135° = 45°
Slide 9-14
Types of Triangles
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Slide 9-15
Similar Figures
„
Two polygons are similar if their corresponding
angles have the same measure and their
corresponding sides are in proportion.
9
6
4
4
3
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6
6
4.5
Slide 9-16
Example
„
Catherine Johnson wants to measure the height
of a lighthouse. Catherine is 5 feet tall and
determines that when her shadow is 12 feet
long, the shadow of the lighthouse is 75 feet
long. How tall is the lighthouse?
x
5
75
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12
Slide 9-17
Example continued
ht. lighthouse lighthouse's shadow
=
ht. Catherine Catherine's shadow
x 75
=
5 12
12 x = 375
x = 31.25
x
5
75
12
Therefore, the lighthouse is 31.25 feet tall.
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Slide 9-18
Congruent Figures
„
„
If corresponding sides of two similar figures are
the same length, the figures are congruent.
Corresponding angles of congruent figures have
the same measure.
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Slide 9-19
Quadrilaterals
„
„
Quadrilaterals are four-sided polygons, the sum
of whose interior angles is 360°.
Quadrilaterals may be classified according to
their characteristics.
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Slide 9-20
Classifications
„
„
Trapezoid
Two sides are parallel.
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„
Parallelogram
„
Both pairs of opposite
sides are parallel. Both
pairs of opposite sides
are equal in length.
Slide 9-21
Classifications continued
„
Rhombus
„
Rectangle
„
Both pairs of opposite sides
are parallel. The four sides are
equal in length.
„
Both pairs of opposite sides
are parallel. Both pairs of
opposite sides are equal in
length. The angles are right
angles.
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Slide 9-22
Classifications continued
„
Square
„
Both pairs of opposite sides
are parallel. The four sides are
equal in length. The angles
are right angles.
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Slide 9-23
9.3
Perimeter and Area
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Formulas
Figure
Rectangle
Square
Parallelogram
Triangle
Trapezoid
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Perimeter
Area
P = 2l + 2w
A = lw
P = 4s
A = s2
P = 2b + 2w
A = bh
P = s1 + s2 + s3
A = 21 bh
P = s1 + s2 + b1 + b2
A = 21 h(b1 + b2 )
Slide 9-25
Example
„
„
„
„
Marcus Sanderson needs to put a new roof on his barn.
One square of roofing covers 100 ft2 and costs $32.00
per square. If one side of the barn roof measures 50
feet by 30 feet, determine
a) the area of the entire roof.
b) how many squares of roofing he needs.
c) the cost of putting on the roof.
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Slide 9-26
Example continued
„
a) The area of the roof is
‰
„
A = lw
A = 30(50)
A = 1500 ft2
1500(2 both sides of the roof) = 3000 ft2
b) Determine the number of squares
area of roof
3000
=
= 30
area of one square 100
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Slide 9-27
Example continued
„
c) Determine the cost
‰
30 squares × $32 per square
$960
‰
It will cost a total of $960 to roof the barn.
‰
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Slide 9-28
Pythagorean Theorem
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Slide 9-29
Example
„
Tomas is bringing his boat into a dock that is 12 feet
above the water level. If a 38 foot rope is attached to the
dock on one side and to the boat on the other side,
determine the horizontal distance from the dock to the
boat.
12 ft
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38 ft rope
Slide 9-30
Example continued
„
a2 + b2 = c 2
122 + b 2 = 383
144 + b 2 = 1444
b 2 = 1300
b = 1300
b ≈ 36.06
2
„
The distance is approximately 36.06 feet.
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Slide 9-31
Circles
„
„
„
A circle is a set of points equidistant from a fixed point
called the center.
A radius, r, of a circle is a line segment from the center
of the circle to any point on the circle.
A diameter, d, of a circle
is a line segment through
the center of the circle with
both end points on the circle.
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Slide 9-32
Example
„
Terri is installing a new circular swimming pool
in her backyard. The pool has a diameter of 27
feet. How much area will the pool take up in her
yard?
A = πr
2
A = π (13.5)2
A = 572.54
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The radius of the pool is
13.5 feet.
The pool will take up about
573 square feet.
Slide 9-33
9.4
Volume
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Volume
„
Volume is the measure of the capacity of a
figure.
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Slide 9-35
Formulas
Figure
Formula
Rectangular Solid
V = lwh
Cube
V = s3
Cylinder
V = πr2h
Cone
V = 31 π r 2 h
Sphere
V = 34 π r 3
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Diagram
Slide 9-36
Example
„
Mr. Stoller needs to order potting soil for his
horticulture class. The class is going to plant
seeds in rectangular planters that are 12 inches
long, 8 inches wide and 3 inches deep. If the
class is going to fill 500 planters, how many
cubic inches of soil are needed?
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Slide 9-37
Example continued
„
We need to find the volume of one planter.
V = lwh
V = 12(8)(3)
V = 288 in.3
„
Soil for 500 planters would be
‰
500(288) = 144,000 cubic inches
‰
144,000
= 83.33 ft 3
1728
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Slide 9-38
Polyhedron
„
A polyhedron is a closed surface formed by the
union of polygonal regions.
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Slide 9-39
Euler’s Polyhedron Formula
„
„
„
Number of vertices − number of edges + number of
faces = 2
Example: A certain polyhedron has 12 edges and 6
faces. Determine the number of vertices on this
polyhedron.
Number of vertices − number of edges + number of
faces = 2
x − 12 + 6 = 2
There are 8 vertices.
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x −6 = 2
x =8
Slide 9-40
Volume of a Prism
„
„
V = Bh, where B is the area of the base and h is
the height.
Example: Find the volume of the figure.
‰
Area of one triangle.
Find the volume.
A = 21 bh
A = 21 (6)(4)
V = Bh
V = 12(8)
4m
A = 12 m2
8m
V = 96 m3
6m
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Slide 9-41
Volume of a Pyramid
„
V = 31 Bh where B is the area of the base and h
„
is the height.
Example: Find the volume of the pyramid.
‰
Base area = 122 = 144
‰
V = 31 Bh
V = 31 (144)(18)
18 m
V = 864 m3
12 m
12 m
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Slide 9-42
9.5
Transformational Geometry,
Symmetry, and Tessellations
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Definitions
„
The act of moving a geometric figure from some
starting position to some ending position without
altering its shape or size is called a rigid
motion (or transformation).
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Slide 9-44
Reflection
„
A reflection is a rigid motion that moves a a
geometric figure to a new position such that the
figure in the new position is a mirror image of
the figure starting position. In two dimensions
the figure and its mirror image are equidistant
from a line called the reflection line or the axis
of reflection.
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Slide 9-45
Construct the reflection of triangle ABC
about the line l.
A
C
A
C
B
l
2 units
B
2 units
B’
l
C’
A’
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Slide 9-46
Translation
„
A translation (or glide) is a rigid motion that
moves a geometric figure by sliding it along a
straight line segment in the plane. The direction
and length of the line segment completely
determine the translation.
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Slide 9-47
Example
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Slide 9-48
Example continued
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Slide 9-49
Rotation
„
A rotation is a rigid motion performed by
rotating a geometric figure in the plane about a
specific point, called the rotation point or the
center of rotation. The angle through which
the object is rotated is called the angle of
rotation.
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Slide 9-50
Example
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Slide 9-51
Example continued
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Slide 9-52
Glide Reflection
„
A glide reflection is a rigid motion formed by
performing a translation (or glide) followed by a
reflection.
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Slide 9-53
Example
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Slide 9-54
Symmetry
„
A symmetry of a geometric figure is a rigid
motion that moves a figure back onto itself. That
is, the beginning position and ending position of
the figure must be identical.
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Slide 9-55
Example
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Slide 9-56
Tessellations
„
A tessellation (or tiling) is a pattern of the
repeated use of the same geometric figures to
entirely cover a plane, leaving no gaps. The
geometric figures use are called the
tessellating shapes of the tessellation.
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Slide 9-57
Example
„
The simplest tessellations use one single
regular polygon.
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Slide 9-58
Example continued
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Slide 9-59