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!
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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.
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Example 1:
Example 2:
Example 3: Match each of the following
1) Rotation
a)
2) Translation
b)
3) Line Reflection
c)
4) Dilation
d)
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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.
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Reflections in the Coordinate Plane
1.
3.
4.
5
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?
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Example 2:
Construct the segment that represents the line of reflection for quadrilateral
What is true about each point on
and its corresponding point on
and its image
.
?
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.
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Example 5
The task at hand is to construct the reflection of
construction on your own.
over line
. Follow the steps below to get started, then complete the
1.
Construct circle : center , with radius such that the circle crosses
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
4.
Connect
,
, and
and
to locate
and
at two points (labeled
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
.
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Reflecting Over Vertical and Horizontal Lines
1.
2.
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Identifying a Reflection
Write a rule to describe each transformation.
1.
2.
Challenge
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Summary
Exit Ticket
Homework:
Homework for Reflections – Day 1
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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.
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Graphing Reflections
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Identifying Reflections
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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.
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5.
Draw a triangle
. Draw a line through vertex
vertex. Construct the reflection across .
so that it intersects the triangle at more than just the
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Translations – Day 2
Do Now:
1.
2.
A translation is a rigid motion. A translation is a direct
isometry. It preserves distance and orientation.
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a.
b. What is the magnitude of the vector?
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Example 5:
a) Name the vector and write its component form.
b) What is the magnitude of the vector?
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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
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4.
5.
6. a) Write a motion rule for the translation.
b) What is the magnitude of the vector?
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7.
Describe the composition of transformations shown in the diagram.
8.
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Homework
1.
2.
3. a) Write a motion rule for the translation. b) What is the magnitude of the vector?
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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.
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Rotations – Day 3
Warm - Up
1.
2.
_______________________________________________________________
A rotation is a rigid motion.
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1a.
___________________________________________________________________
Rules
(x, y)
R90 = ( ___, ____)
R180 = ( ___, ____)
R270 = ( ___, ____)
Practice
3.
(2, 3)
4.
(-1, 6)
5.
(-7, -4)
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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’(
)
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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
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 .
and
.
.
2. Find the center of rotation:
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Identifying a Rotation
Write a rule to describe each transformation.
a.
.
b
Challenge:
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Homework:
1.
Graph the image of the figure under the transformation given and write the coordinates of the
image.
a)
b)
c)
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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.
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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.
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Dilations/Symmetry – Day 4
Warm – Up
1.
2.
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******CHOOSE AN EXPLANATION
A dilation preserves:
A dilation is NOT a rigid motion.
*It does not preserve distance and is not an
isometry.
1)
2)
3)
4)
5)
Angle measure
Parallelism
Colinearity
Midpoint
orientation
34
35
36
Example: Use a compass and straightedge to construct the dilation of
scale factor 3.
LMN with the given center and
37
Homework:
1.
2.
3.
4.
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5.
Graph B(-5,-10), C(-5,0) and D(0,5) and its image after a dilation of -1/5.
6.
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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
scale factor ½ .
LMN with the given center and
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9. Use a compass and straightedge to construct the dilation of
scale factor 2.
LMN with the given center and
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42
Tell whether each figure has rotational symmetry. If so, give the angle of rotational
symmetry.
43
Tell whether each figure has rotational symmetry. If so, give the angle of rotational
symmetry.
44
Summary:
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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.
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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
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Example 2:
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Example 3:
49
Example 4:
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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’’?
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Example 6:
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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?
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5.
6.
55
7.
8.
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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)
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10. Find the angle of rotation that maps A to A’’.
11.
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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 2 10 . What is the length of A’B’, the image of AB after a dilation with a
scale factor of 3?
a) 5 10
5.
b) 2 30
c) 6 10
d ) 6 30
Which transformation is represented in the table below?
a) T-1,3
b) T1,3
c) T-3,1
d) T1,-3
Input
(3,5)
(1,0)
(6,3)
Output
(4,2)
(2,-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) R90o
c)R180o d) rx-axis
9. Which of the following composition of transformations
maps ABC to A’B’C’?
1) R180oo T-2,2
2) T1,4 o ry-axis
3) T3,-2 o rx-axis
4) R90o o ry=x
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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.
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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 R180oo 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’’.
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Lesson
Day 1 – Reflections
Day 2 – Translations
Day 3 – Rotations
Day 4 – Dilations/Symmetry
Day 5 – Compositions and Rigid
Motions
Day 6 – Test Prep Questions
Pages in Packet
1 – 16
17 - 24
25 – 32
33 -45
46 - 58
Homework Assignment
12- 16
23-24
30 – 32
38 – 41 and page 44
54 - 58
59 - 61
59 -61
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