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Properties of Transformations.

Goal: By the end of the week, I will recognize the difference between

translations, reflections, and rotations.

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Transformations: A Real World Connection

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Rigid Transformations

• Translations, rotations, and reflections are rigid transformations.

• Rigid transformations do not change their shape or size. They preserve the distance between any two corresponding points after a transformation has occurred.

• Rigid transformations are congruent. Since the pre-image and image do not change their shape or size, they remain exactly equal in size and shape.

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When you are on an amusement park ride, you are undergoing a transformation.

• A transformation is a change in a figure’s position or size.

• Translations, rotations, and reflections are types of transformations. The original figure is called the pre-image, and the resulting figure is called the image.

• The image of a translation, rotation, or reflection is congruent to the pre-image.

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Labeling Transformations• We label a point on an image by using the

same letter as the corresponding point on the pre-image figure followed by a prime symbol__`__.

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TranslationsA translation "slides" an object a fixed distance in a given direction.  The original object and its translation have the same shape and size, and they face in the same direction.

Translations are SLIDES

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Additional Example 1: Graphing Translations on a Coordinate Plane

Graph the translation of triangle ABC 2 units right and 3 units down.

Add 2 to the x-coordinate of each vertex, and subtract 3 from the y-coordinate of each vertex.

Rule Image

A(–3, 4)A’ (–3 + 2, 4 – 3) A’(–1, 1)

B(0, 2)B’ (0 + 2, 2 – 3) B’(2, –1)

C(–2, 1)C’ (–2 + 2, 1 – 3) C’(0, –2)

A’

B’C’

Transformations7-7Check It Out: Example 1

Graph the translation of the quadrilateral ABCD 3 units down and 5 units left.

Subtract 5 from the x-coordinate of each vertex, and subtract 3 from the y-coordinate of each vertex.

Rule Image

A(1, 4)A’ (1 – 5, 4 – 3) A’(–4, 1)

B(4, 3)B’ (4 – 5, 3 – 3) B’(–1, 0)

C(4, –1)C’ (4 – 5, –1 – 3) C’(–1, –4)

C(1, –2)D’ (1 – 5, –2 – 3) D’(–4, –5)

B’A’

C’D’

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Let’s Practice Translations!!!

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Pre-Image (original) Vertices

Image Vertices

A A’

B B’

C C’

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Pre-Image (original) Vertices

Image Vertices

D D’

E E’

F F’

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Pre-Image (original) Vertices

Image Vertices

G G’

H H’

I I’

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Pre-Image (original) Vertices

Image Vertices

J J’

K K’

L L’

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Exit TicketOn a half sheet of paper, draw a simple figure. Name the pre-image. Now draw the figure being translated and re-name your image using prime notation. In the form of a sentence, tell me how your figure was translated. See example below:

A

A’

I translated figure ABCD by moving it down and to the left.

B’

D

C

B

D’

C’

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A reflection flips a figure across a line to create a mirror image.

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Type Rule

Across the y-axis

Flip the sign of the x-coordinate.(x,y) (-x,y)

Across the x-axis

Flip the sign of the y-coordinate.(x,y) (x,-y)

*Add to guided notes: Across the origin – Flip the sign of

both the x- and y-coordinates.

Transformations7-7Additional Example 2: Graphing Reflections on a

Coordinate Plane

Graph the reflection of quadrilateral ABCD across the y-axis.

Flip the x-coordinate of each vertex.

Rule Image

A(–4, 1)A’ (–1 –4, 1) A’(4, 1)

B(–2, 1)B’ (–1 –2, 1) B’(2, 1)

C(–1, –2)C’ (–1 –1, –2) C’(1, –2)

D(–4, –3)D’ (–1 –4, –3) D’(4, –3)

A’B’

C’D’

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Check It Out: Example 2

Graph the reflection of triangle FGH across the x-axis.

Flip the y-coordinate of each vertex.

Rule Image

F(–4, –2)F’ (–4, –2 –1) F’(–4, 2)

G(1, –3) G’ (1, –3 –1) G’(1, 3)

H(–2, –4)H’ (–2, –4 –1) H’(–2, 4)

H’

G’

F’

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Let’s Practice Reflections!!!

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Pre-Image (original) Vertices

(in black)

Image Vertices(in gray)

A A’

B B’

C C’

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Pre-Image (original) Vertices

(in black)

Image Vertices(in gray)

A A’

B B’

C C’

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Pre-Image (original) Vertices

(in black)

Image Vertices(in gray)

E E’

F F’

G G’

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Pre-Image (original) Vertices

(in black)

Image Vertices(in gray)

E E’

F F’

G G’

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A rotation turns a figure around a point, called the center of rotation.

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ROTATIONS AROUND THE ORIGIN

TYPE RULE

180 degrees Flip the sign of both coordinates.

90 degrees clockwise

negative rotation

Flip the sign of the x-coordinate; then switch the x- and y-coordinates.   

    (x,y) (y,-x)

90 degrees counterclockwise

positive rotation

Flip the sign of the y-coordinate; then switch the x- and y-coordinates.  

(x,y) (-y,x)

Transformations7-7Additional Example 3: Graphing Rotations on a

Coordinate Plane

Graph the rotation of triangle ABC 90 counterclockwise about the origin.

Flip the y-coordinate of each vertex, and switch the x and y coordinates.

Rule Image

A(4, 4)A’ (–1 4, 4 ) A’(–4, 4)

B(4, 1)B’ (–1 1, 4) B’(–1, 4)

C(2, 1)C’ (–1 1, 2) C’(–1, 2)

A’ B’

C’

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Check It Out: Example 3

Graph the rotation of triangle XYZ 180 about the origin.

Flip both the x- and y-coordinates.

Rule Image

X(–1, 2)X’ (–1 –1, –1 2 ) X’(1, –2)

Y(2, 3)Y’ (–1 2, –1 3) Y’(–2, –3)

Z(3, 0)Z’ (–1 3, –1 0) Z’(–3, 0)

Z’

Y’X’

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Let’s Practice Rotations!!!

Transformations7-7Pre-Image (original) Vertices

(in black)

Image Vertices(in gray)

A A’

B B’

C C’

D D’

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(in black)

Image Vertices(in gray)

A A’

B B’

C C’

D D’

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(in black)

Image Vertices(in gray)

L L’

M M’

N N

O O’

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(in black)

Image Vertices(in gray)

L L’

M M’

N N

O O’

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Let’s PracticeWork you examples on your guided notes along

with me.

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1. Graph the translation 3 units right and 4 units down.

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2. Graph the reflection across the x-axis.

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3. Graph the rotation around the origin 90 degrees clockwise.

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What type of transformation has occurred from the original figure to figures 1, 2, and 3?

(1) Translation - the figure was translated up and to the right.

(2) Reflection – the figure was reflected (flipped) down.

(3) Rotation – the figure has been rotated 90-degrees counterclockwise.

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1. Give the coordinates of (1, 4) after a translation 3 units up.

A. (1, 4)

B. (1, 7)

C. (–4, –4)

D. (–4, –7)

Lesson Quiz for Student Response Systems

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2. Give the coordinates of (1, 4) after a reflection across the x-axis.

A. (1, 4)

B. (–1, –4)

C. (1, –4)

D. (–1, 4)

Lesson Quiz for Student Response Systems

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3. Give the coordinates of (1, 4) after a 90 clockwise rotation around the origin.

A. (4, 1)

B. (4, –1)

C. (1, –4)

D. (–4, 1)

Lesson Quiz for Student Response Systems

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More Properties of Transformations

1.Lines to lines

2.Points to points

3.Segments to segments

4.Angles to angles

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Student Outcomes• Students perform translations of figures

along a specific vector. Label the figure using appropriate notation.

• Students learn that a translation maps lines to lines, rays to rays, segments to segments, and angles to angles.

• Students learn that translations preserve lengths of segments and degrees of angles.

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Vocabulary1. Vector: a vector is a directed line

segment, meaning it is a segment with direction given by connecting one of its endpoint (starting point) to the other endpoint. It is often represented as an “arrow” with a “tail.”

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Let’s Translate a Vector• We are going to translate a

vector AB using a transparency.

• Place your transparency over the vector. Trace point P and vector AB exactly as it is. Use precision.

• Keeping the paper fixed in place, slide your transparency along vector AB until the starting point is on top of the endpoint.

• The distance must be the same.

• You have now translated a vector.

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Watch me translate!

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Your diagram shows figures and their images under a translation along vector HI. Trace the vector using your transparency and map the pre-image to its image. Make sure you line up your starting point and map until you hit the endpoint.

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Discussion• A translation maps lines to lines, segments to

segments, angles to angles, and points to points. Does your transparency confirm it?

• A translation preserves the lengths of segments. Did you see that? Was segment DE the same before and after being translated?

• Translations are rigid motions. They did not change the size or shape of the figures. Would you agree?

• Did your translation map angle to angle? Did it preserve the degree of the angle or did it change?

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In Summary…

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Translating Lines

Student Outcomes• Students learn that when lines are

translated, they are either parallel to the given line or they coincide.

• Students learn that translations map parallel lines to parallel lines.

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Vocabulary

1. Coincide: Two lines or shapes that lie directly on top of one another. Two lines coincide when they are on top of each other. Only one line is visible.

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You can only draw one line through Point P that is parallel to Line L.

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The two lines coincide. They are on top of each other.

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The two lines are parallel. They are the same distance apart and will never meet.

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Small GroupsDiscuss the next two problems on your guided notes within your small groups. You have 5 minutes to discuss. Be prepared to share your findings.

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Small Groups

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Small Groups

L1 and L2 are parallel. If we were to translate L1 and L2 along vector DE,

what conclusion could we make about the translated images?

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Summary• We know that there exists just one line,

parallel to a given line and through a given point not on the line.

• We know that translations map parallel lines to parallel lines.

• We know that when lines are translated, they are either parallel to the given line or they coincide.