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New Jersey Center for Teaching and Learning

Progressive Mathematics Initiative

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8th Grade Math

2D Geometry: Transformations

www.njctl.org

2013-06-25




Setting the PowerPoint ViewUse Normal View for the Interactive ElementsTo use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible:

• On the View menu, select Normal.

• Close the Slides tab on the left.

• In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen.

• On the View menu, confirm that Ruler is deselected.

• On the View tab, click Fit to Window.

Use Slide Show View to Administer Assessment ItemsTo administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 21 for an example.)

Table of Contents

• Reflections

• Dilations

• Translations Click on a topic to go to that section

• Rotations

• Transformations

• Congruence & Similarity

Common Core Standards: 8.G.1, 8.G.2, 8.G.3, 8.G.4, 8.G.5

• Special Pairs of Angles

• Symmetry

Transformations

Return to Table of Contents

Any time you move, shrink, or enlarge a figure you make a transformation. If the figure you are moving (pre-image) is labeled with letters A, B, and C, you can label the points on the transformed image (image) with the same letters and the prime sign.

AB

C

A'B'

C'

pre-image image

There are four types of transformations in this unit:

• Translations• Rotations• Reflections• Dilations

The first three transformations preserve the size and shape of the figure.

In other words:If your pre-image is a trapezoid, your image is a congruent trapezoid.

If your pre-image is an angle, your image is an angle with the same measure.

If your pre-image contains parallel lines, your image contains parallel lines.

Translations


A translation is a slide that moves a figure to a different position (left, right, up or down) without changing its size or shape and without flipping or turning it.

You can use a slide arrow to show the direction and distance of the movement.

This shows a translation of pre-image ABC to image A'B'C'. Each point in the pre-image was moved right 7 and up 4.

Click for web page

http://www.mathwarehouse.com/transformations/translations-interactive-activity.php

http://www.mathwarehouse.com/transformations/translations-interactive-activity.php

Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved?Both the pre-image and image are congruent.

A B

CD

A' B'

C'D'

To complete a translation, move each point of the pre-image and label the new point.

Example: Move the figure left 2 units and up 5 units. What are the coordinates of the pre-image and image?

Translate pre-image ABC 2 left and 6 down. What are the coordinates of the image and pre-image?

A

B

C


Translate pre-image ABCD 4 right and 1 down. What are the coordinates of the image and pre-image?

A

B

C

D


A

B

C

D

Translate pre-image ABCD 5 left and 3 up. What is the rule and what are the new coordinates of the image.


A rule can be written to describe translations on the coordinate plane. Look at the following rules and coordinates to see if you can find a pattern.

2 Left and 5 UpA (3,-1) A' (1,4)B (8,-1) B' (6,4)C (7,-3) C' (5,2)D (2, -4) D' (0,1)

2 Left and 6 DownA (-2,7) A' (-4,1)B (-3,1) B' (-5,-5)C (-6,3) C' (-8,-3)

4 Right and 1 DownA (-5,4) A' (-1,3)B (-1,2) B' (3,1)C (-4,-2) C' (0,-3)D (-6, 1) D' (-2,0)

5 Left and 3 UpA (3,2) A' (-2,5)B (7,1) B' (2,4)C (4,0) C' (-1,3)D (2,-2) D' (-3,1)

Translating left/right changes the x-coordinate.

Translating up/down changes the y-coordinate.





Translating left/right changes the x-coordinate.• Left subtracts from the x-coordinate

• Right adds to the x-coordinate

Translating up/down changes the y-coordinate.• Down subtracts from the y-coordinate

• Up adds to the y-coordinate

2 units Left … x-coordinate - 2 y-coordinate stays

rule = (x - 2, y)

5 units Right & 3 units Down… x-coordinate + 5 y-coordinate - 3 rule = (x + 5, y - 3)

click to reveal

A rule can be written to describe translations on the coordinate plane.

click to reveal

Write a rule for each translation.





(x, y) (x-2, y+5) (x, y) (x-2, y-6)

(x, y) (x-5, y+3) (x, y) (x+4, y-1)

click to reveal click to reveal

click to reveal click to reveal

1 What rule describes the translation shown?

D

E

F

G

D'E'

F'

G'

(x,y) (x - 4, y - 6)

(x,y) (x - 6, y - 4)

(x,y) (x + 6, y + 4)

A

B

C

D (x,y) (x + 4, y + 6)


D

E

F

G

D'E'

F'

G'

(x,y) (x, y - 9)

(x,y) (x, y - 3)

(x,y) (x - 9, y)

A

B

C

D (x,y) (x - 3, y)


D

E

F

G

D'

E'

F'

G'

(x,y) (x + 8, y - 5)

(x,y) (x - 5, y - 1)

(x,y) (x + 5, y - 8)

A

B

C

D (x,y) (x - 8, y + 5)


DE

F

G

D'E'

F'

G'

(x,y) (x - 3, y + 2)

(x,y) (x + 3, y - 2)

(x,y) (x + 2, y - 3)

A

B

C

D (x,y) (x - 2, y + 3)


D

E

F

G

D'E'

F'

G'

(x,y) (x - 3, y + 2)

(x,y) (x + 3, y - 2)

(x,y) (x + 2, y - 3)

A

B

C

D (x,y) (x - 2, y + 3)

Rotations


A rotation (turn) moves a figure around a point. This point can be on the figure or it can be some other point. This point is called the point of rotation.

P

Rotation

The person's finger is the point of rotation for each figure.

When you rotate a figure, you can describe the rotation by giving the direction (clockwise or counterclockwise) and the angle that the figure is rotated around the point of rotation. Rotations are counterclockwise unless you are told otherwise.

A

This figure is rotated 90 degrees

counterclockwise about point A.

B

This figure is rotated 180 degrees clockwise

about point B.

Click for answer Click for answer

A B

CD

A'

B' C'

D'

How is this figure rotated about the origin?

In a coordinate plane, each quadrant represents 90.

This figure is rotated 270 degrees clockwise about the origin or 90 degrees counterclockwise about the origin.

Click to Reveal

Check to see if the pre-image and image are congruent.

The following descriptions describe the same rotation. What do you notice? Can you give your own example?

The sum of the two rotations (clockwise and counterclockwise) is 360 degrees. If you have one rotation, you can calculate the other by subtracting from 360.

6 How is this figure rotated about point A? (Choose more than one answer.)

A, A'C'

C

BB'

D' E'

D

E


clockwise

counterclockwise

90 degrees

A

B

C

D 180 degrees

E 270 degrees

7 How is this figure rotated about point the origin? (Choose more than one answer.)

A B

CD

A'B'

C' D'


clockwise

counterclockwise

90 degrees

A

B

C

D 180 degrees

E 270 degrees

8 What is the turn that will turn this hexagon onto itself?

A

B

C

D

When rotated 90 counter-clockwise, the x-coordinate is the opposite of the pre-image y-coordinate and the y-coordinate is the same as the pre-image of the x-coordinate. In other words:

(x, y) (-y, x)

Click to Reveal

A B

CD

A'

B'C'

D'

Now let's look at the same figure and see what happens to the coordinates when we rotate a figure.

Write the coordinates for the pre-image and image.

What do you notice?

When rotated a half-turn, the x-coordinate is the opposite of the pre-image x-coordinate and the y-coordinate is the opposite of the pre-image of the y-coordinate. In other words:

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

Click to Reveal

What happens to the coordinates in a half-turn?

Write the coordinates for the pre-image and image.

What do you notice?

A B

CD

A'B'

C' D'

Can you summarize what happens to the coordinates during a rotation?

90 Counterclockwise:

Half-turn:

90 Clockwise:

(x, y) (-y, x)

(x, y) (y, -x)

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

Click to Reveal

Click to Reveal

Click to Reveal

9 What are the new coordinates of a point A (5, -6) after a 90 rotation clockwise?

(-6, -5)

(-6, 5) (-5, 6)

A

B

C

D (5, -6)

10 What are the new coordinates of a point S (-8, -1) after a 90 rotation counterclockwise?

(-1, -8)

(1, -8)

(-1, 8)

A

B

C

D (8, 1)

11 What are the new coordinates of a point H (-5, 4) after a 180 rotation counterclockwise?

(-5, -4)

(5, -4)

(4, -5)

A

B

C

D (-4, 5)

12 What are the new coordinates of a point R (-4, -2) after a 270 rotation clockwise?

(4, -2)

(-2, 4)

(2, 4)

A

B

C

D (-4, 2)

13 What are the new coordinates of a point Y (9, -12) after a Half-turn?

(-12, 9)

(-9,12)

(-12, -9)

A

B

C

D (9,12)

Reflections


Example

A reflection (flip) creates a mirror image of a figure.

A reflection is a flip because the figure is flipped over a line. Each point in the image is the same distance from the line as the original point.

A

B C

A'

B'C'

t A and A' are both 6 units from line t.B and B' are both 6 units from line t.C and C' are both 3 units from line t.

Each vertex in ABC is the same distance from line t as the vertices in A'B'C'.


Reflect the figure across the y-axis.


x

y

A B

CD

What do you notice about the coordinates when you reflect across the y-axis?

A'B'

C'D'

A (-6, 5) A' (6, 5)B (-4, 5) B' (4, 5)C (-4, 1) C' (4, 1)D (-6, 3) D' (6, 3)

When you reflect across the y-axis, the x-coordinate becomes the opposite.

So (x, y) (-x, y) when you reflect across the y-axis.


Tap box for coordinates

x

y

A B

CD

What do you predict about the coordinates when you reflect across the x-axis?

A' B'

C'D'

A (-6, 5) A' (-6, -5)B (-4, 5) B' (-4, -5)C (-4, 1) C' (-4, -1)D (-6, 3) D' (-6, -3)

When you reflect across the x-axis, the y-coordinate becomes the opposite.

So (x, y) (x, -y) when you reflect across the x-axis.


Tap box for coordinates

Reflect the figure across the y-axis then the x-axis. Click to see each reflection.


Reflect the figure across the y-axis.Click to see reflection.


Reflect the figure across the line x = -2.


Reflect the figure across the line y = x.


14 The reflection below represents a reflection across:

the x axis

the y axis

the x axis, then the y axis

the y axis, then the x axis

x

y

A

B C

A'

B' C'


A B

C D

15 The reflection below represents a reflection across:

x

D

B C

A'

C' B'

D'


the x axis

the y axis

the x axis, then the y axis

the y axis, then the y axis

y

A B

C D

A

16 Which of the following represents a single reflection of Figure 1?

Figure 1

A

B

C

D

17 Which of the following describes the movement below?

reflection

rotation, 90 clockwiseslide

A

B

C

D rotation, 180 clockwise

Describe the reflection below:

x

y

A

B CA'

C'

B'

D'

E'

DE


across the line y = x across the line y = -3

across the y axis across the x axis

A

B

C

D

18

19 Describe the reflection below:

x

y

A

B

C

A'

C'

B'


across the line y = x

across the x axis

across the line y = -3

across the line x = 4

A B

C D

Dilations


The scale factor is the ratio of sides:

When the scale factor of a dilation is greater than 1, the dilation is an enlargement.

When the scale factor of a dilation is less than 1, the dilation is a reduction.

When the scale factor is |1|, the dilation is an identity.

x

y

Example.

If the pre-image is red and the image is blue, what type of dilation is this? What is the scale factor of the dilation?

This is an enlargement.Scale Factor is 2.

x

y

What happened to the coordinates with a scale factor of 2?

AA' B

B'

C C'D D'

A (0, 1) A' (0, 2)B (3, 2) B' (6, 4)C (4, 0) C' (8, 0)D (1, 0) D' (2, 0)

The coordinates were all multiplied by 2.

The center for this dilation was the origin (0,0).

20 What is the scale factor for the image shown below? The pre-image is in red and the image is in blue.

x

y2

3

-3

A

B

C

D 4

21 What are the coordinates of a point S (3, -2) after a dilation with a scale factor of 4 about the origin?

(12, -8)

(-12, -8)

(-12, 8)

A

B

C

D (-3/4, 1/2)

22 What are the coordinates of a point Y (-2, 5) after a dilation with a scale factor of 2.5?

(-0.8, 2)

(-5, 12.5)

(0.8, -2)

A

B

C

D (5, -12.5)

23 What are the coordinates of a point X (4, -8) after a dilation with a scale factor of 0.5?

(-8, 16)

(8, -16)

(-2, 4)

A

B

C

D (2, -4)

24 The coordinates of a point change as follows during a dilation:

(-6, 3) (-2, 1)

What is the scale factor?

3

-3

1/3

A

B

C

D -1/3


(4, -9) (16, -36)


4

-4

1/4

A

B

C

D -1/4


(5, -2) (17.5, -7)


3

-3.75

-3.5

A

B

C

D 3.5

27 Which of the following figures represents a rotation? (and could not have been achieved only using a reflection)

A Figure A B Figure B

C Figure C D Figure D

28 Which of the following figures represents a reflection?



29 Which of the following figures represents a dilation?



30 Which of the following figures represents a translation?



Symmetry


Symmetry

A line of symmetry divides a figure into two parts that match each other exactly when you fold along the dotted line. Draw the lines of symmetry for each figure below if they exist.

Which of these figures have symmetry?Draw the lines of symmetry.

Do these images have symmetry? Where?

Rotational symmetry is when a figure can be rotated around a point onto itself in less than a 360 turn.

Rotational symmetry is when a figure can be rotated around a point onto itself in less than a 360 turn. Rotate these figures. What degree of rotational symmetry do each of these figures have?

31 How many lines of symmetry does this figure have?

3

6

5

A

B

C

D 4

32 Which figure's dotted line shows a line of symmetry?

A B C D

33 Which of the objects does not have rotational symmetry?

A

B

C

D

Rotational symmetry is when a figure can be rotated around a point onto itself in less than a 360 turn.

Click for hint.

Congruence &Similarity


Congruence and Similarity

Congruent shapes have the same size and shape.

2 figures are congruent if the second figure can be obtained from the first by a series of translations, reflections, and/or rotations.

Remember - translations, reflections and rotations preserve image size and shape.

Similar shapes have the same shape, congruent angles and

proportional sides.

2 figures are similar if the second figure can be obtained from the first by a series of translations, reflections, rotations and/or dilations.

Click for web page

http://www.nctm.org/standards/content.aspx?id=26770

http://www.nctm.org/standards/content.aspx?id=26770

Click to reveal

Which pair of shapes is similar but not congruent?

A

B

C

D

34

Which pair of shapes is similar but not congruent?

A

B

C

D

35

Which of the following terms best describes the pair of figures?

congruent

similar

neither congruent nor similar

A

B

C

36

37 Which of the following terms best describes the pair of figures?

congruent

similar


A

B

C

38 Which of the following terms best describes the pair of figures?

congruent

similar


A

B

C

Determine if the two figures are congruent, similar or neither.

Be able to explain how one figure was obtained from the other through a series of translations, rotations, reflections and/or dilations. The pre-image is red, the image is blue.

Special Pairs of Angles

Return to Table Of Contents

Recall:

• Complementary Angles are two angles with a sum of 90 degrees.

These two angles are complementary angles because their sum is 90.

Notice that they form a right angle when placed together.

• Supplementary Angles are two angles with a sum of 180 degrees.

These two angles are supplementary angles because their sum is 180.

Notice that they form a straight angle when placed together.

Vertical Angles are two angles that are opposite each other when two lines intersect.

ab

cd

In this example, the vertical angles are:

Vertical angles have the same measurement. So:

Using what you know about complementary, supplementary and vertical angles, find the measure of the missing angles.

bc

a

By Vertical Angles: By Supplementary Angles:

Click Click

39 Are angles 2 and 4 vertical angles?

12

34

Yes

No

40 Are angles 2 and 3 vertical angles?

12

3

4

Yes

No

41 If angle 1 is 60 degrees, what is the measure of angle 3? You must be able to explain why.

21 3

4

30o

60o

120o

A

B

C

D 15o

If angle 1 is 60 degrees, what is the measure of angle 2? You must be able to explain why.

21 3

4

A

B

C

D

42

30o

60o

120o

15o

Adjacent Angles are two angles that are next to each other and have a common ray between them. This means that they are on the same plane and they share no internal points.

A

B

C

D

ABC is adjacent to CBD

How do you know?• They have a common side (ray )• They have a common vertex (point B)

35

23

Adjacent or Not Adjacent? You Decide!

a

ba

b

a

b

Adjacent Not Adjacent Not Adjacentclick to reveal click to reveal click to reveal

43 Which two angles are adjacent to each other?

1

23

456

1 and 4

2 and 4

A

B

44 Which two angles are adjacent to each other?

12

34 5

6

3 and 6

5 and 4

A

B

Interactive Activity-Click Here

A

PQ

RB

A

E

F

A transversal is a line that cuts across two or more (usually parallel) lines.

http://www.mathopenref.com/transversal.html

Corresponding Angles are on the same side of the transversal and on the same side of the given lines.

a bc d

e f

g h

In this diagram the corresponding angles are:

Tran

sver

sal

Click

45 Which are pairs of corresponding angles?

1 2

3 4

5 6

7 8

2 and 6

3 and 7

1 and 8

A

B

C


1

23

4

5

6

78

2 and 6

3 and 7

1 and 8

A

B

C


1 2

3 4

5 6

7 8

1 and 5

2 and 8

4 and 8

A

B

C

48 Which are pairs of corresponding angles

1

2

3

4 5

6

7

8

2 and 4

6 and 5

7 and 8

A

B

C

D 1 and 3

Alternate Exterior Angles are on opposite sides of the transversal and on the outside of the given lines.

a bc d

e f

g h

In this diagram the alternate exterior angles are:

l

m

n

Which line is the transversal?

Click

Alternate Interior Angles are on opposite sides of the transversal and on the inside of the given lines.

a b

c d

e f

g h

In this diagram the alternate interior angles are:

m

n

l

Click

Same Side Interior Angles are on same sides of the transversal and on the inside of the given lines.

a bc

d

e f

g h

In this diagram the same side interior angles are:

m

n

l

Click

49 Are angles 2 and 7 alternate exterior angles?

Yes

No

1 3

5 7

2 4

6 8

m

n

l


Yes

No

1 3

5 7

2 4

6 8

m

n

l

52 Which angle corresponds to angle 5?

A

B

C

D1 3

5 7

2 4

6 8

m

n

l

3

42

6

53 Which pair of angles are same side interior?

1 3

5 7

2 4

6 8

m

n

l

A

B

C

D

3, 4

4, 7

2, 46, 1

54 What type of angles are and ?

1 3

5 7

2 4

6 8

m

n

l

Alternate Interior Angles

Alternate Exterior Angles

Corresponding Angles

A

B

C

D Vertical Angles

E Same Side Interior

3 6





A

B

C

D Vertical Angles

E Same Side Interior 1 3

5 7

2 4

6 8

m

n

l

5 2





A

B

C

D Vertical Angles

E Same Side Interior

1 3

5 7

2 4

6 8

m

n

l

5 6

57 Are angles 5 and 2 alternate interior angles?

1 3

5 7

2 4

6 8

m

n

lYes

No


Yes

No

1 3

5 7

2 4

6 8

m

n

l

Special Case!!!

If parallel lines are cut by a transversal then:

• Corresponding Angles are congruent• Alternate Interior Angles are congruent• Alternate Exterior Angles are congruent• Same Side Interior Angles are supplementary

SO: 1 3

5 7

2 46 8

l

m

n

are supplementary

are supplementary

click

These Special Cases can further be explained using the transformations of reflections and translations

61 Given the measure of one angle, find the measures of as many angles as possible.Which angles are congruent to the given angle?

4 5

6

2 7

1 8

l

m

n

<4, <5, <6

<5, <7, <1

<2

A

B

C

D <5, <1

62 Given the measure of one angle, find the measures of as many angles as possible.What are the measures of angles 4, 6, 2 and 8?

4 5

6

2 71 8

l

m

n

50o

40o

130o

A

C

B

63 Given the measure of one angle, find the measures of as many angles as possible.Which angles are congruent to the given angle?

1 3

5 7

2 4

8

m

n

l<4

<4, <5, <3

<2

A

B

C

D <8

64 Given the measure of one angle, find the measures of as many angles as possible.What are the measures of angles 2, 4 and 8 respectively?

1 3

5 7

2 4

8

m

n

l

35o, 35

o, 35

o

145o, 35

o, 145

oC

55o, 35

o, 55

0A

B

Applying what we've learned to prove some interesting math facts...

We can use what we've learned to establish some interesting information about triangles.

For example, the sum of the angles of a triangle = 180.

Let's see why!

Given ABC

B

A C

Let's draw a line through B parallel to AC.We then have a two parallel lines cut by a transversal.Number the angles and use what you know to prove the sum of the measures of the angles equals 180.

l

m

n p

B

A C

1

2

l

m

n p

B

A C

1

2

1. since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent.

2. is supplementary with since if 2 parallel lines are cut by a transversal, then same side interior angles are supplementary.

3. Therefore,

1. and since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent.

2. Since all three angles form a straight line, the sum of the angles is 180.

l

m

n p

B

A C

12

Let's look at this another way...

Let's prove the Exterior Angle Theorem -

The measure of the exterior angle of a triangle is equal to the sum of the remote interior angles.

B

A C1

Exterior Angle

Remote Interior Angles

ABC

Let's draw a line through B parallel to AC.We then have a two parallel lines cut by a transversal.Number the angles and use what you know to prove the measure of angle 1 = the sum of the measures of angles B and C.

l

m

n p

B

A C1

2


2.


4. Therefore,

l

m

n p

B

A C1

3

2

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