copyright seventh edition and expanded seventh edition · angles the measure of an angle is the...

Copyright © 2005 Pearson Education, Inc.

Chapter 9

Geometry


9.1

Points, Lines, Planes, and Angles

4Copyright © 2005 Pearson Education, Inc.

Basic Terms

A line segment is part of a line between two points, including the endpoints.

Line segment AB

Ray BA

Ray AB

Line AB

SymbolDiagramDescription

ABsuur

ABuuur

BAuuur

AB

A B

A

A

A

B

B

B


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.


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°


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.


Example

If are supplementary and the measure of ABC is 6 times larger than CBD, determine the measure of each angle.

and ABDABC

A B

C

D180

6 1807 180

25.7

m ABC m CBDx x

xx

+ =

+ =

=

=

o

o

o

o

154.225.7

m ABCm ABD

=

=

o

o


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.


Special Names

5 6

1 24

87

3

One interior and one exterior angles on the same side of the transversal-have the same measure

Corresponding angles

Exterior angles on the opposite sides of the transversal—have the same measure

Alternate exterior angles

Interior angles on the opposite side of the transversal—have the same measure

Alternate interior angles

5 6

1 24

87

3

5 6

1 24

87

3


9.2

Polygons


Polygons

Polygons are names according to their number of sides.

Icosagon20Heptagon7

Dodecagon12Hexagon6

Decagon10Pentagon5Nonagon9Quadrilateral4

Octagon8Triangle3

NameNumber of Sides

NameNumber of Sides


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.


Triangles continued

Determine the sum of the interior angles.

The measure of one interior angle is

The exterior angle is supplementary to the interior angle, so180° − 135° = 45°

( 2)180(8 2)(180 )6(180 )1080

S n= −

= −

=

=

o

o

o

o

1080 1358

=o

o


Types of Triangles


Similar Figures

Two polygons are similar if their corresponding angles have the same measure and their corresponding sides are in proportion.

4

3

4

6

6 6

9

4.5


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

7512

5


Example continued

x

7512

5

ht. lighthouse lighthouse's shadow=ht. Catherine Catherine's shadow

x 755 12

12 37531.25

xx

=

==

Therefore, the lighthouse is 31.25 feet tall.


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.


Quadrilaterals

Quadrilaterals are four-sided polygons, the sum of whose interior angles is 360°.Quadrilaterals may be classified according to their characteristics.


Classifications

Trapezoid

Two sides are parallel.

Parallelogram

Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length.


Classifications continued

Rhombus

Both pairs of opposite sides are parallel. The four sides are equal in length.

Rectangle

Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. The angles are right angles.


Classifications continued

Square

Both pairs of opposite sides are parallel. The four sides are equal in length. The angles are right angles.


9.3

Perimeter and Area


Formulas

P = s1 + s2 + b1 + b2

P = s1 + s2 + s3

P = 2b + 2w

P = 4s

P = 2l + 2w

Perimeter

Trapezoid

Triangle

A = bhParallelogram

A = s2Square

A = lwRectangle

AreaFigure

12A bh=

11 22 ( )A h b b= +


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, determinea) the area of the entire roof.b) how many squares of roofing he needs.c) the cost of putting on the roof.


Example continued

a) The area of the roof isA = lwA = 30(50)A = 1500 ft2

1500(2 both sides of the roof) = 3000 ft2

b) Determine the number of squares

area of roof 3000 30area of one square 100

= =


Example continued

c) Determine the cost30 squares × $32 per square$960

It will cost a total of $960 to roof the barn.


Pythagorean Theorem


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 ft38 ft rope


Example continued

The distance is approximately 36.06 feet.

2 2 2

2 2 3

2

2

2

12 38144 1444

1300

130036.06

a b cbbb

bb

+ =

+ =

+ =

=

=≈


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.


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?

2

2(13.5)572.54

A rAA

ππ

=

==

The radius of the pool is 13.5 feet.

The pool will take up about 573 square feet.


9.4

Volume


Volume

Volume is the measure of the capacity of a figure.


Formulas

Sphere

Cone

V = πr2hCylinder

V = s3Cube

V = lwhRectangular Solid

DiagramFormulaFigure

213V r hπ=

343V rπ=


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?


Example continued

We need to find the volume of one planter.

Soil for 500 planters would be500(288) = 144,000 cubic inches

3

12(8)(3)288 in.

V lwhVV

==

=

3144,000 83.33 ft1728

=


Polyhedron

A polyhedron is a closed surface formed by the union of polygonal regions.


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

There are 8 vertices.

12 6 26 2

8

xx

x

− + =− =

=


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.

8 m

6 m

4 m

1212

2

(6)(4)

12 m

A bhA

A

=

=

= 3

12(8)96 m

V BhVV

==

=


Volume of a Pyramid

where B is the area of the base and his the height. Example: Find the volume of the pyramid.

Base area = 122 = 144

13V Bh=

12 m

12 m

18 m

1313

3

(144)(18)

864 m

V BhV

V

=

=

=


9.5

Transformational Geometry, Symmetry, and Tessellations


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).


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.


Construct the reflection of triangle ABCabout the line l.

A

BC

l l

A

BC

B’

A’

C’

2 units

2 units


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.


Example


Example continued


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.


Example


Example continued


Glide Reflection

A glide reflection is a rigid motion formed by performing a translation (or glide) followed by a reflection.


Example


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.


Example


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.


Example

The simplest tessellations use one single regular polygon.


Example continued

copyright seventh edition and expanded seventh edition · angles the measure of an angle is the...

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