1 chapter 7 transformations. by moving all the points of a geometric figure according to certain...

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Chapter 7

Transformations

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Transformations By moving all the points of a geometric figure according

to certain rules you can create an image of the original figure.

Each point on the original figure corresponds to a point on its image. • Ex: Point A after transformation of any type is called

point A’ (read as “A prime”) If image is congruent to the original figure, the process is

called rigid transformation, or isometry. Transformation not preserving size and shape is called

nonrigid transformation.

Renaming Transformations

It is common practice to name

shapes using capital letters:

It is common practice to name

transformed shapes using the same letters with a

“prime” symbol:

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Types of Transformations

Reflections: These are like mirror images as seen across a line or a point.

Translations ( or slides): This moves the figure to a new location with no change to the looks of the figure.

Rotations: This turns the figure clockwise or counter-clockwise but doesn’t change the figure.

Dilations: This reduces or enlarges the figure to a similar figure.

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Reflections(type of rigid transformation)

You could fold the picture along line l and the left figure would coincide with the corresponding parts of right figure.

l

You can reflect a figure using a line or a point. All measures (lines and angles) are preserved but in a mirror image.

Example: The figure is reflected across line l .

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Reflections – continued…

reflects across the y axis to line n

(2, 1) (-2, 1) & (5, 4) (-5, 4)

Reflection across the x-axis: the x values stay the same and the y values change sign. (x , y) (x, -y)

Reflection across the y-axis: the y values stay the same and the x values change sign. (x , y) (-x, y)

-8 -6 -4 -2 2 4 6 8

-8

-6

-4

-2

2

4

6

8

x

y

Example: In this figure, line l :

reflects across the x axis to line m.

(2, 1) (2, -1) & (5, 4) (5, -4)

ln

m

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Reflections across specific lines:

To reflect a figure across the line y = a or x = a, mark the corresponding points equidistant from the line.

i.e. If a point is 2 units above the line its corresponding image point must be 2 points below the line.

(-3, 6) (-3, -4)

(-6, 2) (-6, 0)

(2, 3) (2, -1).

Example:

Reflect the fig. across the line y = 1.

The line (where a mirror may be placed) is called the line of reflection.  The distance from a point to the line of reflection is the same as the distance from the point's image to the line of reflection.

A reflection can be thought of as a "flipping" of an object over the line of reflection.

If you folded the two shapes together line of reflection the two shapes would overlap exactly!

What happens to points in a Reflection?

Name the points of the original triangle.

Name the points of the reflected triangle.

What is the line of reflection?

How did the points change from the original to the reflection?

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Lines of Symmetry If a line can be drawn through a figure so the one side of the

figure is a reflection of the other side, the line is called a “line of symmetry.”

Some figures have 1 or more lines of symmetry. Some have no lines of symmetry.

One line of symmetry

Infinite lines of symmetry

Four lines of symmetry

Two lines of symmetry

No lines of symmetry

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Translations (slides)(type of rigid transformation)

If a figure is simply moved to another location without change to its shape or direction, it is called a translation (or slide).

A translation also has a particular direction. So you can use a translation vector to describe the translation.

If a point is moved “a” units to the right and “b” units up, then the translated point will be at (x + a, y + b).

If a point is moved “a” units to the left and “b” units down, then the translated point will be at (x - a, y - b).

A

B

Image A translates to image B by moving to the right 3 units and down 8 units.

Example:

A (2, 5) B (2+3, 5-8) B (5, -3)

Let's examine some

translations related to coordinate geometry.

The example shows how each vertex moves the same distance in

the same direction.

Translations

Write the Points

What are the coordinates for A, B, C?

What are the coordinates for A’, B’. C’?

How are they alike? How are they

different?

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Composite Reflections

If an image is reflected over a line and then that image is reflected over a parallel line (called a composite reflection), it results in a translation.

A B C

Image A reflects to image B, which then reflects to image C. Image C is a translation of image A

Example:

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Rotations

An image can be rotated about a fixed point.The blades of a fan rotate about a fixed point.

An image can be rotated over two intersecting lines by using composite reflections.

Image A reflects over line m to B, image B reflects over line n to C. Image C is a rotation of image A.

A

B

C

mn

The concept of rotations can be

seen in wallpaper

designs, fabrics, and art work.

            Rotations are TURNS!!!

Rotations

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RotationsIt is a type of transformation where the object is rotated around a fixed

point called the point of rotation.

When a figure is rotated 90° counterclockwise about the origin, switch each coordinate and multiply the first coordinate by -1. (x, y) (-y, x)

When a figure is rotated 180° about the origin, multiply both coordinates by -1.

(x, y) (-x, -y)

Ex: (1,2) (-1,-2) & (6,2) (-6, -2)

Ex: (1,2) (-2,1) & (6,2) (-2, 6)

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Angles of rotation In a given rotation, where A is the figure and B is the resulting

figure after rotation, and X is the center of the rotation, the measure of the angle of rotation AXB is twice the measure of the angle formed by the intersecting lines of reflection.

Example: Given segment AB to be rotated over lines l and m, which intersect to form a 35° angle. Find the rotation image segment KR.

A

B

35 °

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Angles of Rotation . . Since the angle formed by the lines is 35°, the angle of rotation is

70°. 1. Draw AXK so that its measure is 70° and AX = XK. 2. Draw BXR to measure 70° and BX = XR. 3. Connect K to R to form the rotation image of segment AB.

A

B

35 °X

K

R

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Dilations(type of non-rigid transformation)

A dilation is a transformation which changes the size of a figure but not its shape. This is called a similarity transformation.

Since a dilation changes figures proportionately, it has a scale factor k.• If the absolute value of k is greater than 1, the dilation is an

enlargement.• If the absolute value of k is between 0 and 1, the dilation is a

reduction. • If the absolute value of k is equal to 0, the dilation is

congruence transformation. (No size change occurs.)

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Dilations – continued… In the figure, the center is C. The distance from C to E is three times the distance

from C to A. The distance from C to F is three times the distance from C to B. This shows a transformation of segment AB with center C and a scale factor of 3 to the enlarged segment EF.

In this figure, the distance from C to R is ½ the distance from C to A. The distance from C to W is ½ the distance from C to B. This is a transformation of segment AB with center C and a scale factor of ½ to the reduced segment RW.

C

E

A

FB

C

R

A

BW

A dilation used to create an image larger than the original is called an enlargement.  A

dilation used to create an image smaller than the original is called a reduction.

Dilation

Dilations always involve a change in size.

Notice how EVERY

coordinate of the original triangle has

been multiplied by

the scale factor (x2).

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Dilations – examples…Find the measure of the dilation image of segment AB, 6 units long,

with a scale factor of 1. S.F. = -4: the dilation image will be an enlargment since the

absolute value of the scale factor is greater than 1. The image will be 24 units long.

2. S.F. = 2/3: since the scale factor is between 0 and 1, the image will be a reduction. The image will be 2/3 times 6 or 4 units long.

3. S.F. = 1: since the scale factor is 1, this will be a congruence transformation. The image will be the same length as the original segment, 1 unit long.

REVIEW: Answer each question………………………..

Does this picture show a translation,

rotation, dilation, or reflection?

How do you know?

Rotation

Does this picture show a

translation, rotation, dilation,

or reflection?

How do you know?

Dilation

Does this picture show a translation, rotation, dilation, or

reflection?

How do you know?

(Line) Reflection

Which of the following lettered figures are translations of the

shape of the purple arrow?  Name ALL that apply.

Explain your thinking.

Letters a, c, and e are translations of the purple

arrow.

Has each picture been rotated in a clockwise or counter-clockwise

direction?

The birds were rotated clockwise and the fish

counterclockwise.

Dutch graphic artist M. C. Escher (1898-1972) is known for his creative use of tessellations in his work. What transformations can you see in this

picture?

The birds and fish have been translated here.

What transformations can you see in this Escher print?

Some birds have been translated and some have been rotated.

Name 5 examples in real life of each transformation. Include pictures. Due

next class. We will show and tell.

• Translation

• Rotation

• Reflection

• Dilation