Transformations

A transformation is a process that manipulates a polygon or other two-dimensional object on a plane or coordinate system.

​​MGSE8.G.1 Verify experimentally the congruence properties of rotations, reflections, and translations: lines are taken to lines and line segments to line segments of the same length; angles are taken to angles of the same measure; parallel lines are taken to parallel lines.

MGSE8.G.2 Understand that a two‐dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

MGSE8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two‐ dimensional figures using coordinates.

MGSE8.G.4 Understand that a two‐dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two‐dimensional figures, describe a sequence that exhibits the similarity between them.

What is a Transformation

Take notes.

Multiple Choice

The original figure prior to a transformation.

• Pre-image

• Image

Multiple Choice

The result of a transformation.

• Pre-image

• Image

Rotation = ​Turn

Reflection =Flip

Translation = Slide

Dilation = ​Enlarge/Reduce

​

tHERE ARE 4 TYPES OF trNASFORMATIONS

Multiple Choice

A slide is also called a ___________.

• Translation

• Reflection

• Rotation

• Transformation

Multiple Choice

Which is the only transformation that changes a shapes size?

• Translatoin

• Rotation

• Reflection

• Dilation

Translations

Multiple Choice

The Transformation in this picture is

• Rigid

• Non Rigid

• We can't tell!

​TRANSFORMATION OF SHAPES CAN BE:

Multiple Choice

Which of the 4 types of transformations is non-rigid transformation?

• Rotation

• Reflection

• Translation

• Dilation

Multiple Choice

Identify the transformation.

• Reflection across y-axis

• 90 Rotation counter clockwise

• Translation 5 units left, 1 unit up.

• Translation 5 units right, 1 unit down

Multiple Choice

How is triangle ABC being translated to triangle A'B'C'?

• up 3, left 8

• up 9, left 4

• down 3, right 8

• down 9, right 4

Multiple Choice

Which rule describe the translation that maps PQRS into P'Q'R'S'?

• $$T_{<8,-2>}$$  

• $$T_{<8,2>}$$  

• $$T_{<-8,-2>}$$  

• $$T_{<-8,2>}$$  

Sliding the shape in the coordinates plane by moving it:

Right (+) or Left (-) (on x Axis)

Up (+) or down​ (-) (on y Axis)

​All points move same distance.

TRanslations

TRanslation Rule

​

In this example we can write the rule as:

Multiple Choice

The translation image of $$\Delta LMN\ is\ \Delta L'M'N'$$  with L'(1,-2), M'(3,-4), N'(6,-2).

Which rule describe the translation?

• $$T_{<-5,1>}=\left(x-5,\ y+1\right)$$  

• $$T_{<1,7>}=\left(x+1,\ y+7\right)$$  

• $$T_{<5,-1>}=\left(x+5,\ y-1\right)$$  

• $$T_{<7,-1>}=\left(x+7,\ y-1\right)$$  

Multiple Select

Choose all of the following that will translate shape B to shape A.

• Horizontal -3

  Vertical -5

• 3 Left

  5 Down

• 2 Left

  4 Up

• Horizontal -2

  Vertical 4

Multiple Select

Choose all of the following that will translate shape A to shape B.

• (x,y) --> (x-5, y-3)

• Horizontal -5

  Vertical -3

• 5 Right

  3 Up

• Horizontal 5

  Vertical 3

Reflections

Multiple Choice

Identify line of reflection

• x axis

• y axis

• y = x

• y = -x

Multiple Choice

Identify line of reflection

• y axis

• x = -4

• y = -4

• y = -x

Multiple Choice

Identify the line of reflection

• y = x

• y = -x

• x axis

• y axis

Multiple Choice

What is the image of R(-5,-4), after a reflection over the line y = x

• R'(-5,-4)

• R'(-4,-5)

• R'(5, 4)

• R'(-4, 5)

The pre-image of the triangle is the blue XYZ

The reflection is the green X'Y'Z'

Each triangle is the same number of units from the x-axis (the line of reflection).

​

Reflection of ΔXYZ

Multiple Choice

Which two triangles are reflections of each other across the x-axis?

• A and B

• B and D

• A and C

• C and D

Multiple Choice

Where will A' be at, if triangle ABC is reflected by
(x,y) --> (-x,y)?

• (2,1)

• (1,-2)

• (-1,2)

Multiple Choice

Reflect over x-axis. Which point would be at (3, -6)

• A

• B

• C

• H

Multiple Choice

Find K' if the figure is reflected across the x-axis

• (4, 1))

• (-1, -4)

• (1, -4)

• (1, 4)

Multiple Select

Select all the sets that are reflections across the X AXIS.

• A(5, 6) --> A'(5, -6)

• B(3, -8) --> B'(3, 8)

• C(5, -5) --> C'(-5, -5)

• D(-6, -1) --> D'(-6, 10

• E(1, 6) --> E'(1, -6)

Rotations

Rotations

• Rotation is a "TURN"s

• Rotations are considered RIGID because the pre-images and final images will always be CONGRUENT (Same Shape, Same Size, Same Measurements, Same Lengths)

Multiple Choice

Rotate the point (5,5) around the origin 180 degrees counterclockwise. State the image of the point.

Rule: (x, y) ---- ( -x, -y)

• (5,-5)

• (5,5)

• (-5,5)

• (-5,-5)

Multiple Choice

Which rule describes rotating 180° clockwise?

• ( x, y ) → ( y, -x )

• ( x, y ) → ( -x, -y )

• ( x, y) → ( x, y )

• ( x, y ) → ( -y, x )

Multiple Choice

Rotate the point (-3,-4) around the origin 180 degrees. State the image of the point.

Rule: (x, y) ---- ( -x, -y)

• (-4, -3)

• (-4, 3)

• (4, -3)

• (3, 4)

Multiple Choice

Rotate the point (-5,8) around the origin 90 degrees clockwise. State the image of the point.

Rule: (x, y) ----- (y, -x)

• (-8,5)

• (8,-5)

• (8,5)

• (5,-8)

Multiple Choice

If you were to rotate ABCD 90°clockwise about the origin, what would the coordinate of A' be?

Rule: (x, y) ----- (y, -x)

• (-5, 5)

• (5, -3)

• (-5, 3)

• (-3, 3)

Multiple Choice

I'm (-4, 5) and I moved counterclockwise 1 quadrant. Where am I?

• I

• II

• III

• v

What does (-x, -y) mean?

It means you change the sign of x and change the sign of y.

If x is negative, changes to positive

If x is positive ,changes to negative

If y is negative, changes to positive

If y is positive, changes to negative

Multiple Choice

Triangle B is rotated 90° clockwise with the origin as the center of rotation to create a new figure. Which triangle shows the new location?

• A

• B

• C

• D

Multiple Choice

What is the image of F (-2, 7) after a rotation of 180° clockwise?

• F' (-2, -7)

• F' (7, -2)

• F' (2, -7)

• F' (2, 7)

Multiple Choice

Which rule describes rotating 90° clockwise?

HINT: SAME AS 270 DEGREES COUNTER CLOCKWISE

• ( x, y ) → ( y, -x )

• ( x, y ) → ( -x, -y )

• ( x, y) → ( x, y )

• ( x, y ) → ( -y, x )

Multiple Choice

What is the image of M (4, -8) after a rotation 270° clockwise?

TRICK QUESTION: SAME AS 90 DEGREES COUNTER CLOCKWISE

• M' (-8, 4)

• M' (8, 4)

• M' (-4, -8)

• M' (-8, -4)

Multiple Choice

Triangle ABC is shown on the graph. What are the coordinates of the image of point B after the triangle is rotated 270° about the origin?

• (4, 2)

• (2, 4)

• (–4, –2)

• (–2, –4)

Multiple Choice

What will be the coordinates of the image of ABC after it is rotated 90 degrees counterclockwise about the origin?

• (-4, -3), (-1, 0), (-2, -5)

• (3, -4), (0, -1), (5, -2)

• (-3, 4), (0, 1), (-5, 2)

• (4, 3), (1, 0), (2, 5)

Open Ended

Rotate triangle ABC with vertices A(1,2) B(3,4) C(5,6) 180 degrees.

Dilations Practice

Multiple Choice

Triangle ABC is dilated to produce triangle A'B'C' with scale factor 3/4 . Which describes the relationship between the two triangles?

• Triangle A'B'C' is an enlargement of triangle ABC

• Triangle A'B'C' is a reduction of triangle ABC

• Triangle A'B'C' is a mirror image of triangle ABC

Scale Factor is represented by the letter k

• When k > 1, the shape is enlarged (gets bigger)

• When k < 1, the shape is reduced (get smaller)

• When k = 1, the shape stays the same

Multiple Choice

Which is true about the scale factor when a figure is enlarged?

• The scale factor is zero.

• The scale factor is 1.

• The scale factor is larger than 1.

• The scale factor is smaller than 1.

Now Let's Practice more Dilation Rules

• Remember: dilation rules are the ONLY rule where x and y are MULTIPLIED by a coefficient

• Dilation rules will always look like this, where k is the scale factor:

• (x, y) --> (kx, ky)

Multiple Choice

• F

• G

• H

• J

Multiple Choice

• A

• B

• C

• D

Multiple Choice

• A

• B

• C

• D

Multiple Choice

An image is dilated at a scale factor of 1.5. This means the dilation is an enlargement and the dilated image will be bigger.

• True

• False

Multiple Choice

Dilate the figure by a scale factor of 3 with the origin as the center of dilation. What are the coordinates of the image?

A(3,3) B(6,6) C(9,3)

• A'(9,9) B'(18,18) C'(27,9)

• A'(6,6) B'(9,9) C'(12,6)

• A'(0,0) B'(3,3) C'(6,0)

• A'(1,1) B'(2,2) C'(3,1)

Multiple Choice

Which of the following describes the sequence of transformations shown?

• Reflect across the x-axis and then rotate -90 degrees cw around the origin

• Rotate 90 degrees cw around the origin and then translate down.

• Reflect across the x-axis and then reflect across the y-axis.

• Translate up and then rotate 90 degrees ccw about the origin.

Multiple Choice

What sequence of transformations maps the red parallelogram to the green parallelogram?

• A translation followed by a reflection.

• A rotation followed by a translation.

• A reflection followed by a translation.

• A reflection followed by a rotation.

Multiple Choice

Which sequence could be used to transform hexagon C to hexagon D?

• Translation 4 left and 2 down

• a reflection over the x-axis and a translation 6 units right

• a reflection over the y-axis and a translation 2 units down

• a counter-clockwise rotation of 90 degrees about the origin

Multiple Choice

What sequence of transformations could map triangle ABC to triangle A"B"C"?

• A dilation followed by a rotation.

• A translation followed by a reflection.

• A rotation about the origin followed by a translation.

• A reflection followed by a rotation.

Multiple Choice

Which of the following describes the sequence of transformations shown?

• Reflection followed by a rotation.

• Rotation followed by a translation.

• Reflection across the x-axis followed by a reflection across the y-axis.

• Translation followed by a rotation.