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# Chapter 9

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by

## Josh Truax

on 22 February 2018

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#### Transcript of Chapter 9

A transformation maps every point of a figure onto its image and may be described with arrow (-->) notation.

Prime Notation ( ' ) -
used to identify image points
Section 9.1 - Translation
Transformation -
results in a change in the position, shape, or size of a figure.

Preimage -
the original figure
Image -
the figure after transformation
Section 9.2 - Reflection
Reflection (aka: Flip) -
an isometry in which a figure and its image have opposite orientations. Think of how a reflected image in a mirror appears "backwards"
Section 9.3 - Dilation
Dilation -
a transformation whose preimage and image are similar.
Since a dilation involves a change in size, it is
NOT
an isometry
Section 9.4 - Symmetry
Symmetry -
exists within a figure if there is an isometry that maps the figure onto itself
Section 9.5 - Compositions of Isometries
Isometry -
same distance
Isometry only exists in transformations that preserve distance or length
This means that translations and reflections are isometries.

**Dilation is not an isometry**
Section 9.6 - Tessellation
Tessellation
Also known as "tiling", it is a repeating pattern of figures that completely covers an area without gaps or overlaps

Chapter 9
Transformations

Translation -
a transformation that maps all of the points of a figure the same distance in the same direction.

You write a translation as:
T (ΔABC) = ΔA'B'C'
Write a rule to describe the below translation: T(PQRS) = P'Q'R'S'
Translate each figure based on the give rule.

D
(6,2)
H
(3, -1)
S
(-2,-4)
->
(x+2, y-2)

W
(2,2)
X
(2,-2)
Y
(-2,-2)
Z
(-2,2)
->
(x, y+4)
Homework #10 - Section 9.1
Complete the assignment on Math XL.

**Remember!!! You must complete 100% of the assignment in order to earn full credit in the Grade book**
Isometry -
occurs when the pre-image and image are congruent
Only exists if there is no change to the size or shape of an object after its transformation
All reflections will have the following properties:
1.
Reflections preserve distance
If point A is on the line of reflection then A = A'
If point B is not on the line of reflection, the the line of reflection is the perpendicular bisector of BB'
2.
Reflections preserve angle measures
3.
Reflections match each point of the preimage to one and only one point on the corresponding image
Reflect the following:

P
across the x-axis

V
across the y-axis

S
across x = 1

U
across y = -1
Draw the following figure in the coordinate plane. Then reflect the preimage across each indicated line of reflection.
D(-3,4) H(0,1) S(2,3)
Reflect across:

The x-axis The y-axis
Homework #11 - Section 9.2
Complete the assignment on Math XL.

**Remember!!! You must complete 100% of the assignment in order to earn full credit in the Grade book**
Every dilation has:
1. A center
2. A scale factor

The scale factor describes the change in size between the preimage and the image
Enlargement -
scale factor > 1
Reduction -
scale factor < 1
Homework #12 - Section 9.3
Complete the assignment on Math XL.

**Remember!!! You must complete 100% of the assignment in order to earn full credit in the Grade book**
Line/Reflectional Symmetry
One half of the figure is a mirror image of the other half. If the figure were folded along the line of symmetry, the two halves would match up exactly
Rotational Symmetry
Occurs when a figure can be rotated some degree up to 180° and still look the same as before the rotation

Point Symmetry -
a figure that has symmetry at exactly 180° rotation
Draw all of the lines of symmetry for a regular hexagon.

Draw all of the lines of symmetry for a rectangle.
Symmetry Project
You have just started your own company and must now create a company logo. This logo must contain
both line symmetry and rotational symmetry.

The company logo must be your own creation and
cannot be a logo that already exists
. Your logo should also have something to do with your company. Once you have drawn your company logo,
provide your company name and a short description of what your company does
.
Theorem 9.1 - Compositions of Isometries
The composition of two or more isometries is an isometry.
There are only four kinds of isometries.
Theorem 9.2 - Reflections Across Parallel Lines
A composition of reflections across two parallel lines is a translation.
AA", BB", and CC" are all perpendicular to lines l and m
Theorem 9.3 - Reflections Across Intersecting Lines
A composition of reflections across two intersecting lines is a rotation.
The figure is rotated about the fixed point Q
Glide Reflections
Glide Reflection
The composition of a translation (glide) and a reflection across a line parallel to the direction of the translation.

Find the image of the letter for two reflections: first across
line a
and again across
line b
.
Is the resulting transformation a translation or a rotation?
Find the image of the letter for a reflection across
line a
and then again across
line b
.
What is the center of rotation?
What is the angle of rotation?
Find the image of triangle TEX for a glide reflection where the
translation is (x, y - 5)
and the line of reflection is the
y-axis
.

T(-5,2) E(-1,3) X(-2,1)
Homework #13 - Section 9.5
Complete the assignment on Math XL.

**Remember!!! You must complete 100% of the assignment in order to earn full credit in the Grade book**
Because the figures in a tessellation do not overlap or leave gaps, the
sum of the measure of the angles
around any vertex
must equal 360°
If the angles around a vertex are all congruent, then the measure of each angle must be a
factor of 360
.
180 (n-2)
n
Determine if the following figures will tessellate.
Theorem 9.4
Every triangle tessellates.
Theorem 9.5
Symmetry in Tessellations
Reflectional Symmetry
Can be reflected around some line and remain the same

Rotational Symmetry
Can be turned around some angle and remain the same

Translational Symmetry
Can be slid some direction and remain the same

Glide Reflectional Symmetry
Can be translated and reflected and remain the same
Determine what type(s) of symmetry the tessellation has.
Determine what type(s) of symmetry the tessellation has.
Tessellation Project
Notebook Question #1
Notebook Question #2
Notebook Question #3
You are visiting San Francisco. From your hotel near Union Square, you walk 4 blocks east and 4 blocks north to the Wells Fargo History Museum. Then you walk 5 blocks west and 3 blocks north to the Cable Car Barn Museum. Where is the Cable Car Barn Museum in relation to your hotel?
Notebook Question #1
Notebook Question #2
Notebook Question #3
When you play pool, you can use the fact that the ball bounces off the side of the pool table at the same angle at which it hits the side. Suppose you want to put the ball at point B into the pocket at point P by bouncing it off side RS. Off what point on RS should the ball bounce? Draw a diagram and explain your reasoning.
Notebook Question #1
Notebook Question #2
Notebook Question #3
A magnifying glass shows you an image of an object that is 7 times the object’s actual size. So the scale factor of the enlargement is 7. The photo shows an apple seed under this magnifying glass. What is the actual length of the apple seed?
Notebook Question #1
Notebook Question #2
Find 3 letters of the alphabet (capitalized) that have line symmetry. Write out the letters and draw the line of symmetry.

Find 3 letters of the alphabet (capitalized) that have point symmetry. Are there any letters that have both?
Notebook Question #1
Notebook Question #2
Notebook Question #3
Notebook Question #1
Notebook Question #2
Notebook Question #3
Full transcript