Chapter 9

Transformations

9.1 Reflections

Types

o There are four types of transformations:o Reflectionso Translationso Rotationso Dilations

o The first three are congruency (Isometry) transformations. In other words the new figure is congruent to the old figure (Pre-Image).

o Dilations are similarity transformations. The new image is different in size from the pre-image.

Reflections

o Key words are “mirror image” or “flip”

o You can reflect across a line or a point.

o For Coordinate Geometryo Most common lines are the y-axis, x-axis,

y = x line or any horizontal or vertical line.o Most common point that the pre-image is

reflected across is the origin.o We will use matrices for coordinate

geometry.

Reflections across a line

Line of ReflectionNotice the new image is a “flip” of the pre-image across the line of reflection.

If we pick two corresponding points on the pre-image and the reflected image, notice the line of reflection is the perpendicular bisector of the segments.

A A’

B B’

Reflections across a line

Line of ReflectionWhat happens when the line of reflection goes through the pre-image?

Notice the new image is a “flip” of the pre-image across the line of reflection.

AA’

C C’B B’

Notice the pre-image C is on the line of reflection, thus the new image C’ is in the exact place.

Reflections across a point

Point of Reflection

Notice the new image is a “flip” of the pre-image across the point of reflection.

It looks very similar to a “rotation.” In 9.3 you will see why.

The point of reflection is the midpoint between any point on the pre-image and the new image.

Coordinate Geometry

o There are specific rules you need to memorize in order to do reflections in coordinate geometry.

o Reflect across x axis – P(x,y) P’(x, -y)

o Reflect across y axis – P(x,y) P’(-x, y)

o Reflect across origin – P(x,y) P’(-x, -y)

o Reflect across y=x line P(x,y) P’(y,x)

o Notice which points become negative!

Example

o Take ΔABC where A(-3, 4), B(0, 8) and C(5, -2)

o Ref across x axis:o A’(-3, -4), B’(0, -8) and C’(5, 2)o y’s change sign.

o Ref across y axis:o A’(3, 4), B’(0, 8) and C’(-5, -2)o x’s change sign.

Example Continued

o Take ΔABC where A(-3, 4), B(0, 8) and C(5, -2)

o Ref across origin:o A(3, -4), B(0, -8) and C(-5, 2)o Everything changes sign.

o Ref across y = x line:o A(4, -3), B(8, 0) and C(-2, 5)o x’s and y’s change position.

Review of Matrices

o Matrices can be added, subtracted, scalar multiplied and multiplied.

o You did the first three in algebra I. You must know the last one for this section.

o The good part is that the TI – 83 does it for you without you having to know how manually.

o The size of the matrix is the number of rows by the number of columns.

Addition of Matrices

o Matrices can only be added or subtracted if they are the same size. That is the same number of row and the same number of columns.

a b c g h i

d e f j k l

a g b h c i

d j e k f l

2x3 2x3 2x3

Scalar Multiplication

o Scalar Multiplication is very similar to distribution. You have a constant outside of the matrix multiplying the matrix by it.

b c dae f g

ab ac ad

ae af ag

Multiplication

o Multiplication is very intricate. All you will need to know how to do is plug the matrices into the calculator and multiply.

o To multiply matrices the number of columns of the first matrix must equal the number of rows in the second matrix.

o You can multiply a 2x3 by a 3x5 because the number of columns in the first (3) is equal to the number or rows in the second (3).

o You can’t do the reverse.

Multiplication

o Matrix [A]=

o Matrix [B]=

o Find [A][B] =

2

3

1

9

3

2 3 4

1 2 5

Multiplication (H)

2

3

1

9

3

2 3 4

1 2 5

(2)(2)+(3)(-3)+(-4)(1) = -9

(-1)(2)+(2)(-3)+(5)(1) = -3

(2x3) by (3x1) = (2x1)

So why Matrices

o If you’re giving coordinates for any polygon you can put those coordinates in matrix form.

o For example ΔABC where A(-3, 4), B(0, 8) and C(5, -2) can be written in a 2x3 size matrix like this:

3 0 5

4 8 2

Point APoint BPoint C

Reflections with Matrixes

1 0

0 1

1 0

0 1

1 0

0 1

o Ref across x axis,multiply by this:

o Ref across y axis,multiply by this:

o Ref across origin,multiply by this:

o Ref across y = x line,Multiply by this:

0 1

1 0

Reflections Across x axis

o All you need to do is multiply the matrix that you will use for the reflection and the matrix that is for the polygon.

o The result will be the new matrix.

1 0

0 1

3 0 5

4 8 2

3 0 5

4 8 2

Reflections with Matrices

o You must remember the order is important!

o The first matrix is the reflection matrix

o The second matrix is the matrix for the polygon.

o If you mix the order of the matrices up, you will not be able to multiply them.

o [A][B] is not always equal to [B][A]….

9.2 Translations

Translations

o The key word for translations is “slide”

o You can translate “slide” a figure along a line.

o The key point is the all corresponding points move the exact same distance.

o This is also a congruency (Isometry) transformation.

Translations

A

A’

B

B’C

C’

Notice all segments are congruent to each other. It is the distance from corresponding points that are all the same.

Compositions

o Compositions are multiple transformations.

o A Translation can be made by double reflections across parallel lines.

Coordinate Geometry

o You can translate (slide) either parallel to the x axis, parallel to the y axis or a do successive translations where you move along the x axis first, then the y axis.

o Example of RULE:o To move a point 5 to the right and 2 down.

P(x,y) P’(x + 5, y – 2)o P(3, 5) o P’( 3 + 5, 5 – 2) o P’ (8, 3)

Matrix Translations

o Matrix Translations are the easiest of all the matrix transformations.

o There is only one matrix to memorize and there is only addition.

o Take Quadrilateral ABCD where A(-2, 5), B(0, 6), C(3, 0) and D(7, -1) and we want to move it 5 units to the right and 2 units down.

o Remember the Rule: P(x,y) P’(x+5, y-2)?

Matrix Translations

2 0 3 7

5 6 0 1

5 5 5 5

2 2 2 2

oQuadrilateral ABCD where A(-2, 5), B(0, 6), C(3, 0) and D(7, -1)

3 5 8 12

3 4 2 3

oOriginal Matrix (Pre-Image)

oTranslation Matrix to move a quadrilateral “5 units to the right and 2 units down”

oTranslated Quadrilateral A’B’C’D’

9.3 Rotations

Rotations

o The third transformation is a Rotation.o The key word is “spin”o You will rotate (Spin) an object about a point.o A rotation is also another Isometry

transformation.o This point is called the “center of rotation”o In coordinate geometry, it usually is the origin.o You will rotate the figure a certain number of

degrees called the “angle of rotation”.

Rotation

Center of Rotation

Rotations can be clockwise or counter clockwise.

Rotations

o Rotations can also be thought of as a composition transformation.

o It is a double reflection across non-parallel lines.

o The angle made between the non-parallel lines is ½ the angle of rotation.

Double Reflection

50°

The 50° angle made between the intersection lines of reflection creates a 100° angle of rotation.

Coordinate Geometry

o Coordinate Geometry rotations are performed with the origin as the center of rotation.

o There are three matrices you need to memorize to do this.

o Remember, when you multiply you must put one of these three matrices first then the matrix for the polygon second.

Coordinate Geometry

0 1

1 0

0 1

1 0

Rotate 90° CCW or 270°CW.

Rotate 270° CCW or 90°CW.

Rotate 180° CCW or 180°CW. 1 0

0 1

This is the same matrix as reflecting across a point.

Example

2 0 3 7

5 6 4 1

0 1

1 0

Rotate Quadrilateral ABCD 90 degrees clockwise, where A(-2, 5), B(0, 6), C(3, 4) and D(7, -1)

5 6 4 1

2 0 3 7

Example

6

4

2

-2

-4

-6

-8

-5 5

Example6

4

2

-2

-4

-6

-8

-5 5

90° CW rotation around origin.

9.5 Dilations

Dilations

o Dilations are the only transformations that are not Isometry.

o They are similarity transformations.o So, the pre image and the new image are not

the same size or location.o Dilations can be enlargements or reduction

depending on the |r|.If |r|>1 you have an enlargement.If 0 < |r| < 1 you have a reduction.

o You will have a center of dilation.

Example r = 2

Center of Dilation

Since r = 2, the new location of the tail will be twice as far away from the center of dilation as the pre-image.

Same thing for the new location of the eye. It will be twice as far away from the center of dilation as the pre-image.

What you end up with is a figure that is twice as large (b/c r = 2) as the pre-image AND twice as far away from the Center of Dilation.

r = 3

Center of Dilation

What do you think will happen when r = 3?

You have a figure that is 3 times as large and 3 times as far away from the Center as the pre-image.

Example r = 1/2

Center of Dilation

Since r = 1/2, the new location of the tail will be half as far away from the center of dilation as the pre-image.

Same thing for the new location of the eye. It will be half as far away from the center of dilation as the pre-image.

What you end up with is a figure that is half as large (b/c r = 1/2) as the pre-image AND half as far away from the Center of Dilation.

Dilations and Coordinate Geometry

o Dilations are pretty easy with coordinate geometry and matrices.

o You will need to do scalar multiplication of the matrix.

o The scalar multiplier is your r!

r = 2

o Dilate ΔABC where A(-3, 4), B(3, 3) and C(5, -2) with the origin as the center point.

o Since r = 2, you will need to multiply the matrix by 2.

3 3 5

4 3 2

6 6 10

8 6 4

2

Example

-10 -5 5 10

6

4

2

-2

-4

-6

-8

-10 -5 5 10

8

6

4

2

-2

-4

-6

-8

Example