1

Reflections – Day 1

A transformation is a function that moves or changes a figure in some

way to produce a new figure call an image. Another name for the

original figure is the pre-image.

We will study the following transformations:

Reflections

Translations

Rotations

Dilations R Resizing!

2

Vocabulary to know:

A Rigid Motion is a transformation that preserves length (distance

preserving) and angle measure (angle preserving).

Another name for a rigid motion is an isometry.

Orientation refers to the arrangement of points relevant to one another,

after a transformation has occurred.

A direct isometry preserves distance and orientation. Translations and

Rotations are direct isometries.

An opposite isometry preserves distance but does not preserve

orientation. Line reflections are opposite isometries.

3

Example 1:

Example 2:

Example 3: Match each of the following

1) Rotation a)

2) Translation b)

3) Line Reflection c)

4) Dilation d)

4

Line Reflections

Line reflections are isometries that do not preserve orientation.

Therefore line reflections are opposite isometries.

Thus, a reflected image in a mirror appears "backwards."

A line reflection is a rigid motion.

____________________________________________________

Example:

a.

5

Reflections in the Coordinate Plane

1.

3. 4.

6

5. Triangle: A(1,2), B(1,4) and C(3,3) .

Reflection: Origin

A’( )

B’( )

C’( )

Rule: rorigin (x,y) = ( , )

Examples:

1. What is the image of point A(4,-7) after a reflection in the x-axis?

2. What is the image of point B(2,-3) after a reflection in the y-axis?

3. Find the coordinates of C’, the image of C(5,-2) , after a reflection in ry=x.

4. Find the coordinates of the image of point D(-3,7) after a reflection in the line y = -x.

Constructions and Line Reflections.

Example 1:

is reflected across and maps onto .

Use your compass and straightedge to construct the perpendicular

bisector of each of the segments connecting to , to , and to .

What do you notice about these perpendicular bisectors?

Label the point at which intersects as point . What is true

about and ? How do you know this is true?

7

Example 2:

Construct the segment that represents the line of reflection for quadrilateral and its image .

What is true about each point on and its corresponding point on ?

Examples 3–4

Construct the line of reflection across which each image below was reflected.

3. 4.

You have shown that a line of reflection is the perpendicular bisector of segments connecting corresponding points on a figure and

its reflected image. You have also constructed a line of reflection between a figure and its reflected image. Now we need to explore

methods for constructing the reflected image itself. The first few steps are provided for you in this next stage.

8

Example 5

The task at hand is to construct the reflection of over line . Follow the steps below to get started, then complete the

construction on your own.

1. Construct circle : center , with radius such that the circle crosses at two points (labeled and ).

2. Construct circle : center , radius and circle : center , radius . Label the [unlabeled] point of intersection

between circles and as point . This is the reflection of vertex across .

3. Repeat steps 1 and 2 for vertices and to locate and .

4. Connect , , and to construct the reflected triangle.

Things to consider:

When you found the line of reflection earlier, you did this by constructing perpendicular bisectors of segments joining two

corresponding vertices. How does the reflection you constructed above relate to your earlier efforts at finding the line of reflection

itself? Why did the construction above work?

Example 6

Now try a slightly more complex figure. Reflect across line .

9

Reflecting Over Vertical and Horizontal Lines

1.

2.

10

Identifying a Reflection

Write a rule to describe each transformation.

1. 2.

Challenge

11

Summary

Exit Ticket

Homework:

Homework for Reflections – Day 1

12

Homework

1. Find the reflection of the point (5,7) under a reflection in the x-axis.

2. Find the reflection of the point (6,-2) under a reflection in the y-axis.

3. Find the reflection of the point (-1,-6) under a reflection in the line y =x.

4.

5.

13

Graphing Reflections

14

Identifying Reflections

15

Construct the line of reflection for each pair of figures below.

1.

2.

3.

4. Reflect the given image across the line of

reflection provided.

16

5. Draw a triangle . Draw a line through vertex so that it intersects the triangle at more than just the

vertex. Construct the reflection across .

17

Translations – Day 2

Do Now:

1.

2.

A translation is a rigid motion. A translation is a direct

isometry. It preserves distance and orientation.

18

a.

b. What is the magnitude of the vector?

19

Example 5:

a) Name the vector and write its component form.

b) What is the magnitude of the vector?

20

Practice Problems

1. The vertices of ΔLMN are L(2,2), M(5,3) and N(9,1). Translate ΔLMN using the vector <-2, 6 >.

2. Graph ΔRST with vertices R(2,2), S(5,2) and T(3,5) and its image under the translation

(x,y) ( x+ 1, y + 2).

Triangle 3. Δ DEF is a translation of ΔABC. Use the diagram to write a rule for

the translation of ΔABC to ΔDEF.

DEF is a

21

4.

5.

6. a) Write a motion rule for the translation.

b) What is the magnitude of the vector?

22

7. Describe the composition of transformations shown in the diagram.

8.

23

Homework

1.

2.

3. a) Write a motion rule for the translation. b) What is the magnitude of the vector?

24

4. Write a rule for the translation.

5. Find the component form of the vector that translates P(-3,6) to P’(0,1).

6. Describe the composition of transformations shown in the diagram.

7.

25

Rotations – Day 3

Warm - Up

1.

2.

_______________________________________________________________

A rotation is a rigid motion.

26

___________________________________________________________________

Rules

(x, y)

R90 = ( ___, ____)

R180 = ( ___, ____)

R270 = ( ___, ____)

Practice

3. 4. 5.

1a.

(2, 3) (-1, 6) (-7, -4)

27

6. Given triangle X(2,2), Y(4,3) and Z(3,0).

Write the coordinates of its image

after a rotation 900.

Rule: (x,y) → ( )

X’( )

Y’( )

Z’( )

7. Given triangle A(1,3), B(4,4) and C(4,0).

Write the coordinates of its image

after a rotation 1800.

Rule: (x,y) → ( )

A’( )

B’( )

C’( )

8. Given triangle A(-2,-5), B(-1,1)

and C(1,-2).

Write the coordinates of its image

after a rotation 2700.

Rule: (x,y) → ( )

A’( )

B’( )

C’( )

28

Finding the center of rotation using constructions

1. Follow the directions below to locate the center of rotation taking the figure at the top right to its image at the

bottom left.

a. Draw a segment connecting points and .

b. Using a compass and straightedge, find the perpendicular bisector of this segment.

c. Draw a segment connecting points and .

d. Find the perpendicular bisector of this segment.

e. The point of intersection of the two perpendicular bisectors is the

center of rotation. Label this point .

2. Find the center of rotation:

29

Identifying a Rotation Write a rule to describe each transformation.

a. b.

Challenge:

30

Homework:

1. Graph the image of the figure under the transformation given and write the coordinates of the image.

a)

b)

c)

31

2. Write the rule for the transformation.

3. Write the rule for the transformation.

4. Graph the image of triangle DEF after a rotation of 180o.

32

5. Use a compass and a straightedge to find the center of rotation.

6. On your paper, construct an equilateral triangle. Locate the midpoint of one side using your

compass. Rotate the triangle around this midpoint. What figure have you formed?

7. Write the rule for the transformation.

33

Dilations/Symmetry – Day 4

Warm – Up

1. 2.

34

******CHOOSE AN EXPLANATION

A dilation is NOT a rigid motion.

*It does not preserve distance and is not an

isometry.

A dilation preserves:

1) Angle measure

2) Parallelism

3) Colinearity

4) Midpoint

5) orientation

35

36

37

Example: Use a compass and straightedge to construct the dilation of LMN with the given center and

scale factor 3.

38

Homework:

1.

2.

3.

4.

39

5. Graph B(-5,-10), C(-5,0) and D(0,5) and its image after a dilation of -1/5.

6.

40

7. Find the scale factor of the dilation for each of the following: a)

b)

c)

8. Use a compass and straightedge to construct the dilation of LMN with the given center and scale factor ½ .

41

9. Use a compass and straightedge to construct the dilation of LMN with the given center and scale factor 2.

42

43

Tell whether each figure has rotational symmetry. If so, give the angle of rotational

symmetry.

44

Tell whether each figure has rotational symmetry. If so, give the angle of rotational

symmetry.

45

Summary:

46

Compositions and Rigid Motions – Day 5

Warm – Up

1.

2.

Which rotation about point O maps B to D? _______

A composition of transformations is when two or more

transformations are combined to find a single transformation.

The composition of two or more rigid motions is a rigid motion.

47

Example 1:

___________________________________________________________________________________

The symbol for a composition of transformations is an open circle.

The notation is read as a reflection in the x-axis following a

translation of (x+3, y+4). Be careful!!! The process is done in reverse!!

You may see various notations which represent a composition of transformations:

could also be indicated by

48

Example 2:

49

Example 3:

50

Example 4:

51

Example 5: Use the figure. The distance between line k and line m is 1.6

centimeters.

a) The preimage is reflected in line k, then in line m. Describe a single transformation that maps

the figure on the far left to the figure on the far right.

b) What is the relationship between PP’ and line k? Explain.

c) What is the distance between P and P’’?

52

Example 6:

53

54

Homework:

1. Identify any congruent figures in the coordinate plane.

2. Describe a congruence transformation that maps Quadrilateral SPQR to ZWXY.

3. Describe a congruence transformation that maps triangle ABC to triangle EFG.

4. a) Evaluate the composition of rx –axis o R90 (-3,5). b) What single transformation would be the same as the composition in part a?

55

5.

6.

56

7.

8.

57

9. Determine whether the polygons with the given vertices are congruent. Use transformations to explain your reasoning. a)

b) W(-3,1), X(2,1), Y(4,-4), Z(-5,-4) and C(-1,-3), D(-1,2), E(4,4), F(4,-5)

58

10. Find the angle of rotation that maps A to A’’.

11.

59

Test Prep Questions

1. Which transformation preserves both distance and angle measure? a) Translation b) horizontal stretch c) vertical stretch d) dilation

2. Which description is always the composition of two different types of transformations? a) Line reflection b) rotation c) translation d) glide reflection

3. Which of the following transformations preserve angle measure but not distance? a) Dilation b) translation c) vertical stretch d) rotation

4. AB has a length of 102 . What is the length of A’B’, the image of AB after a dilation with a

scale factor of 3?

a) 105 b) 302 c) 106 d ) 306

5. Which transformation is represented in the table below? a) T-1,3 b) T1,3 c) T-3,1 d) T1,-3

Input Output

(3,5) (4,2)

(1,0) (2,-3)

(6,3) (7,0)

6. Which rigid motion does not preserve orientation? a) line reflection b) rotation c) translation d) dilation

7. Which transformation is not considered a rigid motion? a) dilation b) rotation c) reflection d) translation

8. Which transformation is equivalent to a reflection over a point? a) T(2,5) b) R90

o c)R180o d) rx-axis

9. Which of the following composition of transformations

maps ABC to A’B’C’? 1) R180

oo T-2,2 2) T1,4 o ry-axis 3) T3,-2 o rx-axis 4) R90

o o ry=x

60

10. A series of three transformations that, when performed on ABC, finally resulting in A’’’B’’’C’’’. Identify each of the three transformations:

a) ABC to A’B’C’

b) A’B’C’ to A’’B’’C’’

c) A’’B’’C’’ to A’’’B’’’C’’’

d) Is ABC A’’’B’’’C’’’? Justify your answer.

e) Identify a single transformation that, when performed on A’’’B’’’C’’’, would result in a

triangle A’’’B’’’C’’’ such that ABC is not congruent to A’’’B’’’C’’’. Justify your answer.

11. Use a sequence of rigid motions to show that 1 2. Use coordinate notation. Label the vertices in triangle 2 using A’B’C’ to correspond to the appropriate angles in triangle 1.

61

12. a) On the accompanying grid, graph and label XYZ with coordinates X(0,2), Y(5,0) and Z(1,3).

b) On the same grid graph and label X’’Y’’Z’’, the image of XYZ after the composition of transformations R180

oo rx-axis.

c) What single transformation maps XYZ to X’’Y’’Z’’?

13. Given circle A with the equation (x-2)2 + (y + 3)2 = 4. a) Perform the translation of circle A: T(2,-4) and label the translation as circle A’ b) Perform a dilation of circle A’ about its center with a scale factor of 2. Label the image A’’. c) Write the equation of circle A’ and circle A’’.

62

Lesson Pages in Packet Homework Assignment

Day 1 – Reflections 1 – 16 12- 16

Day 2 – Translations 17 - 24 23-24

Day 3 – Rotations 25 – 32 30 – 32

Day 4 – Dilations/Symmetry 33 -45 38 – 41 and page 44

Day 5 – Compositions and Rigid Motions

46 - 58 54 - 58

Day 6 – Test Prep Questions 59 - 61 59 -61