© 2008 Pearson Addison-Wesley. All rights reserved Chapter 10 Modeling with Geometry

Copyright © 2008 Pearson Education, Inc. Slide 9-2 Chapter 10 Modeling with Geometry 10AFundamentals of Geometry 10BProblem Solving with Geometry 10CFractal Geometry

Copyright © 2008 Pearson Education, Inc. Slide 9-3 Unit 10A Fundamentals of Geometry

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 line plane

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

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

Copyright © 2008 Pearson Education, Inc. Slide 9-7 Polygons 10-A

Copyright © 2008 Pearson Education, Inc. Slide 9-8 Perimeter and Area 10-A

Copyright © 2008 Pearson Education, Inc. Slide 9-9 Surface Area and Volume 10-A

Copyright © 2008 Pearson Education, Inc. Slide 9-10 Platonic Solids 10-A

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

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

Copyright © 2008 Pearson Education, Inc. Slide 9-13 Unit 10B Problem Solving with Geometry

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

What is? , Hey Dude, meet me at , Copyright © 2008 Pearson Education, Inc. Slide 9-15

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

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 a 2 + b 2 = c 2 10-B a b c

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

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.

Copyright © 2008 Pearson Education, Inc. Slide 9-20 Unit 10C Fractal Geometry

Copyright © 2008 Pearson Education, Inc. Slide 9-21 A Fractal Landscape by Anne Burns 10-C

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 = R D where N is the factor by which the number of elements increases when we shorten the ruler by a reduction factor R.

Copyright © 2008 Pearson Education, Inc. Slide 9-23 The Snowflake Island 10-C

Copyright © 2008 Pearson Education, Inc. Slide 9-24 Sierpinksi Triangle Iterations 10-C

Copyright © 2008 Pearson Education, Inc. Slide 9-25 Sierpinksi Triangle Iterations 10-C

Copyright © 2008 Pearson Education, Inc. Slide 9-26 Sierpinksi Triangle Iterations 10-C

Copyright © 2008 Pearson Education, Inc. Slide 9-27 Sierpinksi Triangle Iterations 10-C

Copyright © 2008 Pearson Education, Inc. Slide 9-28 The Mandelbrot Set 10-C

Copyright © 2008 Pearson Education, Inc. Slide 9-29 Two Views of Barnsley’s Fern 10-C