© 2008 pearson addison-wesley. all rights reserved chapter 10 modeling with geometry
TRANSCRIPT
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© 2008 Pearson Addison-Wesley.All rights reserved
Chapter 10Modeling with Geometry
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Copyright © 2008 Pearson Education, Inc. Slide 9-2
Chapter 10Modeling with Geometry
10A Fundamentals of Geometry
10B Problem Solving with Geometry
10C Fractal Geometry
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Copyright © 2008 Pearson Education, Inc. Slide 9-3
Unit 10A
Fundamentals of Geometry
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Copyright © 2008 Pearson Education, Inc. Slide 9-4
Points, Lines and Planes
A geometric point is imagined to have zero size. A geometric line is formed by connecting two
points along the shortest possible path. Line segments are pieces of a line A geometric plane is a perfectly flat surface that
has infinite length and width but no thickness.
10-A
point lineplane
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Copyright © 2008 Pearson Education, Inc. Slide 9-5
Angles
The intersection of two lines or line segments
forms an angle.
The point of intersection is called the vertex.
10-A
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Copyright © 2008 Pearson Education, Inc. Slide 9-6
Types of Angles
Right: measures 90 Straight: measures 180
Acute: measures less Obtuse: measures between than 90 90 and 180
10-A
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Copyright © 2008 Pearson Education, Inc. Slide 9-7
Polygons
10-A
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Copyright © 2008 Pearson Education, Inc. Slide 9-8
Perimeter and Area
10-A
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Copyright © 2008 Pearson Education, Inc. Slide 9-9
Surface Area and Volume
10-A
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Copyright © 2008 Pearson Education, Inc. Slide 9-10
Platonic Solids
10-A
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Copyright © 2008 Pearson Education, Inc. Slide 9-11
Scaling Laws
Lengths always scale with the scale factor.
Areas always scale with the square of the scale factor.
Volumes always scale with the cube of the scale factor.
10-A
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Copyright © 2008 Pearson Education, Inc. Slide 9-12
The Surface-Area-to-Volume Ratio
Larger objects have smaller surface-area-to-volume ratios than similarly proportioned small objects
Smaller objects have larger surface-area-to-volume ratios than similarly proportioned small objects
10-A
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Copyright © 2008 Pearson Education, Inc. Slide 9-13
Unit 10B
Problem Solving with Geometry
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Copyright © 2008 Pearson Education, Inc. Slide 9-14
Latitude and Longitude
We can locate any place on the Earth’s surface by its latitude and longitude.
10-B
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What is?
33.931586,-109.444891
Hey Dude, meet me at 33.931586,-109.444891
Copyright © 2008 Pearson Education, Inc. Slide 9-15
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Copyright © 2008 Pearson Education, Inc. Slide 9-16
Angular Size and Distance
The farther away an object is located from you, the smaller it will appear in angular size.
10-B
angular size physical size
360 2 distancep=
° ´
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Copyright © 2008 Pearson Education, Inc. Slide 9-17
Pythagorean Theorem
The Pythagorean theorem applies only to right triangles (those with one 90 angle). For a right triangle with side lengths a, b, and c, in which c is the longest side (or hypotenuse), the Pythagorean theorem states that
a2 + b2 = c2
10-B
a
b
c
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Copyright © 2008 Pearson Education, Inc. Slide 9-18
Similar Triangles
Two triangles are similar if they have the same shape, but not necessarily the same size, meaning that one is a scaled-up or scaled-down version of the other.
10-B
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Copyright © 2008 Pearson Education, Inc. Slide 9-19
Similar Triangles
10-B
Corresponding pairs of angles in each triangle are equal.
The ratios of the side lengths in the two triangles are all equal.
, ,m A m A m B m B m C m CÐ = Ð Ð = Ð Ð = Т ¢ ¢
a b c
a b c= =
¢ ¢ ¢
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Copyright © 2008 Pearson Education, Inc. Slide 9-20
Unit 10C
Fractal Geometry
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Copyright © 2008 Pearson Education, Inc. Slide 9-21
A Fractal Landscape by Anne Burns
10-C
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Copyright © 2008 Pearson Education, Inc. Slide 9-22
Fractal Dimension & The Snowflake Curve
10-C
The fractal dimension of an object is defined as a number D such that
N = RD
where N is the factor by which the number of elements increases when we shorten the ruler by a reduction factor R.
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Copyright © 2008 Pearson Education, Inc. Slide 9-23
The Snowflake Island
10-C
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Copyright © 2008 Pearson Education, Inc. Slide 9-24
Sierpinksi Triangle Iterations
10-C
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Copyright © 2008 Pearson Education, Inc. Slide 9-25
Sierpinksi Triangle Iterations
10-C
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Copyright © 2008 Pearson Education, Inc. Slide 9-26
Sierpinksi Triangle Iterations
10-C
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Copyright © 2008 Pearson Education, Inc. Slide 9-27
Sierpinksi Triangle Iterations
10-C
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Copyright © 2008 Pearson Education, Inc. Slide 9-28
The Mandelbrot Set
10-C
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Copyright © 2008 Pearson Education, Inc. Slide 9-29
Two Views of Barnsley’s Fern
10-C