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

© 2008 Pearson Addison-Wesley.All rights reserved

Chapter 10Modeling with Geometry

Copyright © 2008 Pearson Education, Inc. Slide 9-2

Chapter 10Modeling with Geometry

10A Fundamentals of Geometry

10B Problem Solving with Geometry

10C Fractal Geometry


Unit 10A

Fundamentals of Geometry


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


Angles

The intersection of two lines or line segments

forms an angle.

The point of intersection is called the vertex.

10-A


Types of Angles

Right: measures 90 Straight: measures 180

Acute: measures less Obtuse: measures between than 90 90 and 180

10-A


Polygons

10-A


Perimeter and Area

10-A


Surface Area and Volume

10-A


Platonic Solids

10-A


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


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


Unit 10B

Problem Solving with Geometry


Latitude and Longitude

We can locate any place on the Earth’s surface by its latitude and longitude.

10-B

What is?

33.931586,-109.444891

Hey Dude, meet me at 33.931586,-109.444891



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=

° ´


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


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


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 = Ð¢ ¢ ¢

a b c

a b c= =

¢ ¢ ¢


Unit 10C

Fractal Geometry


A Fractal Landscape by Anne Burns

10-C


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.


The Snowflake Island

10-C


Sierpinksi Triangle Iterations

10-C



10-C


The Mandelbrot Set

10-C


Two Views of Barnsley’s Fern

10-C

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

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