Chapter 9 Transformations. 9.1 Reflections Types oThere are four types of transformations:...
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Transcript of Chapter 9 Transformations. 9.1 Reflections Types oThere are four types of transformations:...
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