GEOMETRY TRANSFORMATION:ROTATION & DILATION
objective: Students will be able to
will be able to: Identify rotations in a plane., rotate an object on the coordinate plan
Identify dilation , - dilate a shape on the coordinate plane.
• Dilate points on a coordinate plane by a scale factor.
• Describe and identify transformations in the plane, using proper function notation.
• Rotate a geometric figure. • Reflect a figure over a line of symmetry.
• Translate a figure by sliding it to a different location.
•Transform figure using dilations (scale factor) by enlarging or reducing the size of a figure without
changing its form or shape.
identify the type of symmetry in a figure
Compare transformations that preserve distance and angles to those that do not. perform and interpret a given
sequence of transformations and draw the result
Solve problems involving transformations in order to solve real-world problem and identify types of
symmetry in real-life objects.
STANDARDS
Using transformational geometry, create a reflection, translation, rotation, glide reflection and dilation of a
figure; and apply transformations and use symmetry to analyze mathematical situations
Cluster Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid
motions in geometric concepts, e.g., translations move points a specified distance along a line parallel to a
specified line; rotations move objects along a circular arc with a specified center through a specified angle
G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe
transformations as functions that take points in the plane as inputs and give other points as outputs. Compare
transformations that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch)
1
Make geometric constructions
G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.
G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments
G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using,
e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry
a given figure onto another.
ESSENTIAL QUESTIONS
ENDURING
VOCABULARY
MATERIALS
UNDERSTANDING
What does "transform" mean, and
Manipulation of geometric
rotation
geometer's sketchpad
what does it enable us to
figures can
reflection
graph paper
understand?
be a useful tool in real world
dilation
journal
Why is it important to be able to
situations
translation
straight edge
move a shape?
glide reflection
What are the similarities and
image
differences between the images and
original
ruler, patty paper, Mira™
pre-images
rotation reflection
Optional – Dynamic
generated by translations?
translation translation vector
geometry software
What is the relationship between
direction distance
the coordinates of the vertices of a
angle of rotation center of
figure and the
rotation
coordinates of the vertices of the
line of reflection rigid
figure’s image generated by
transformation
translations?
symmetry • reflectional
How can translations be applied to
symmetry • line symmetry
real-world situations?
• rotational symmetry • point
preimage or
Worksheets, protractor,
symmetry
2
STANDARDS FOR MATHEMATICAL PRACTICE
Guiding Benchmark
Think about your students. Is this
learning goal new information? Do
they have the prior knowledge
necessary? Is this challenging?
Content and Task Decisions
RIGOR
What math skills and mathematical
practices will need to be
demonstrated to be successful with
this task?
How do I modify the task to provide
access for struggling or advanced
learners?
Skills and Modifications
Mathematical Practices
-Make sense of problems and persevere
in solving them
-Reason abstractly and quantitatively
-Construct viable arguments and
critique the reasoning of others
-Model with mathematics
-Use appropriate tools strategically
-Attend to precision
-Look for and make use of structure
-Look for and express regularity in
repeated reasoning
Real Life Connection
When would you use this skill?
Student Task
 Problem Based – Keep it
simple
 Engages students in the
intended mathematics/learning
target (conceptual, pictorial,
abstract)
 May have multiple solutions
 Requires students to wrestle
with the main ideas
Student-Oriented: Develop parameters, structure, and focus for the assignment.
Launch
Teaching Actions
Before Phase 5 – 10 minutes
 Get students Mentally
Prepared
 Be sure task is understood
 Establish expectations
Explore
During Phase 15 – 20 minutes
How will you assess student progress
toward learning goals?
What questions will you ask to
scaffold learning?
Summary
After Phase 15 – 20 minutes
Solution Share: Math Talk
How will you document/record
student solutions?
How will you connect student
discoveries to the learning target?
Extension Ideas
What will children do after the
lesson? Opportunities that could
extend this activity.
3
Student Name:
Parent Name
Period ________
Parent Signature:______
WARM UP:
Triangle ABC has vertices A(-2, 3) , B(2, 4) and C(2,0).
1. Find the coordinates of ABC after translating left 3 and down 1.
2. Find the coordinates of ABC after reflecting it over the x-axis
3. Find the coordinates of ABC after reflecting it over y = x
4. Find the coordinates of ABC after reflecting it over the y-axis then translating it right 4 and down 3.
4
Student Name:
Parent Name
Period ________
Parent Signature:______
WARM UP II
Look at the parallelogram on the coordinate plane below.
Complete the following:
• On the grid provided above, reflect ABCD across the y-axis. Label the image A'B'C'D'.
• Use mathematics to explain how a reflection across the y-axis affects the coordinates of any point
x, y . Use words, symbols, or both in your explanation.
• On the same grid provided above , translate A'B'C'D' 6 units down. Label the image of this
translation A"B"C"D".
• Use mathematics to explain how a translation 6 units down affects the coordinates of any point (x, y).
Use words, symbols, or both in your explanation.
5
Introduction: Have students watch http://www.to14.com/game.php?id=4d486a31e395d
Definition: A rotation is a transformation in which a figure is turned about a fixed point.
The fixed point is the center of rotation Rays drawn from the center of rotation to a point and its image form
an angle called the angle of rotation
• Rotations move an object about a central point
• Windmill vanes rotate around a central arm as do the hands of a clock
Rotations can be clockwise or counterclockwise
Rotating a Figure
Use the following steps to draw the image of ABC after a 120° counterclockwise rotation about point
P.
Draw a segment connecting vertex A and the center
of rotation point P
Place the point of the compass
at P and draw an arc from A to
locate A'.
ANGLE
SINE
COSINE
Use a protractor to measure a 120° angle
counterclockwise and draw a ray
Repeat Steps 1–3 for each
vertex. Connect the vertices
to form the image.
COSINE AND SINE OF REGULAR ANGLES OF FIRST QUADRANT




00
30 0  rad
45 0  rad
60 0  rad
90 0  rad
6
4
3
2
0
1
1
2
3
2
2
2
1
0
1
3
2
2
2
2
6
Theorem 1: A rotation is an isometry
ROTATION ABOUT A POINT WHICH IS NOT THE ORIGIN
Rotation is the transformation along circular paths. Rotation angle  determines the amount of rotation for
- sin  
cos 
each vertex. In order to rotate a point around the origin you use the rotation matrix 
cos 
sin 
Let M x, y  be any point on the plane. Then the rotation of the point M about an angle  about the origin
- sin    x 
 x'  cos 
maps it onto a point M ' x' , y' such that    
which is the rotation of the matrix
cos   y 
 y ' sin 
about the origin through an angle  .
Remark: To rotate about a point that is not the origin, first you move all the points so the center is the
origin, use the usual rotation matrix, and then move all the points back to where you found them.
For instance, if the center is (3,5) you first subtract (3,5) from all the coordinates, then use your matrix, then
add the (3,5) back to everything.


Example 1: If M 10, 3 is rotated about an angle 30 0 , determine the image point.
here   30 0 , Cos30 0  
cos 30
sin 30

 3
 x'   2
 y '  
  1

2
3
1
, Sin 30 0   . Rotation matrix is given by
2
2
 3
1
- 

- sin 30
2
 2
Substituting the above values

cos30   1
3


2 
2
1
-  10
2     4.5 3 
 .Therefore, the image point is given by M ' 4.5 3 ,7
  
7 
3   3  

2 


Definition: A figure has symmetry when isometries other than the identity isometry leave it unchanged
while permuting its parts. These are called the symmetry group of the figure. Symmetry means balance or
form.
REFLECTIONAL/ROTATIONAL SYMMETRY
Definition: A figure has symmetry if there is an isometry that maps the figure onto itself. If the isometry
is the reflection of a plane figure, the figure has reflectional symmetry or line symmetry (i.e two halves
match an identical.) .One half of the figure is a mirror image of its other half. Fold the figure along the line of
symmetry and the halves match exactly
A figure that has rotational symmetry is its own image for some rotation of 180 0 or less. A figure that has
point symmetry has 180 0 rotational symmetry. A square has 90 0 and 180 0 rotational symmetry with the
center of rotation at the center of the square. Thus, a square also has point symmetry.
7
ROTATIONAL SYMMETRY
A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180° or
less. For instance, a square has rotational symmetry because it maps onto itself by a rotation of 90°. The
angle of rotation for rotational symmetry is the smallest angle needed for the figure to rotate onto itself.
Take an object turned the object with a certain number of degrees and the object still looks the same, i.e. it
matches itself a number of times while it is being rotated. This type of symmetry is called rotational
symmetry.
A figure is said to posses a rotational symmetry if it fits onto itself more than once while being rotated
through 360◦. If A◦ is the smallest angle by which a particular figure has to be rotated so that its rotated form
360
its fits onto the original form, then the order of rotational symmetry is given by
. For a figure to possess
A
a rotational symmetry, we must have A 0  180 0 . Thus, the order of rotational symmetry of a figure is the
number of times the figure fits onto itself in the process of rotation through 360◦.
Definition:
Rotational symmetry is when we turn an object tit looks exactly the same. The number of positions in which
it looks exactly the same gives you its
order of symmetry.
Example 2: Consider a rectangle rotate this rectangle through 360 0 about the centre point (upon rotation
90◦,180◦, 270◦ and 360◦). Stop the rotation when it comes its first position.
Clearly, the rectangle take 4 steps to take its first position i.e. the rectangle possess a rotational symmetry of
order 4.
Example 3: The equilateral triangle has rotational symmetry. The angle of rotation is 120 0
.
8
Definition: If a line divides a given figure into two coincidental parts, then we say that the figure is
symmetrical about the line and the line is called axis of symmetry or line of symmetry.
(or)
A line that divides a figures into two congruent figures is called line of
symmetry
Example 4: A circle has infinite number of lines of symmetry. Parallelogram do not have line of symmetry.
Letter ‘A’ has vertical line symmetry In this letter two halves match
Letter ‘B, C, D and E’ has horizontal line symmetry. The top part
is exactly coincide the bottom part
F, G, J, , L, N, P, Q, R, S, Z do not have line of symmetry.
9
0
 360 
Theorem 2: a regular n  gon has n lines of symmetry and 
 rotational symmetry.
 n 
Example 5: Which figures have rotational symmetry? For those that do, describe the rotations
that map the figure onto itself.
SOLUTION
a. This octagon has rotational symmetry. It can be mapped onto itself by a clockwise or counterclockwise
rotation of 45°, 90°, 135°, or 180° about its center.
b. This parallelogram has rotational symmetry. It can be mapped onto itself by a clockwise or
counterclockwise rotation of 180° about its center.
c. The trapezoid does not have rotational symmetry.
Example 6: (Connection to Physical Science)Some objects in nature that have symmetry, are insects,
amoebas, starfish, and minerals., Butterfly, Thajmahal, bow, human body
Example 7:
For a regular hexagon, There are 6
lines of symmetry that make
mirror-image congruent halves.
for a rectangle, there are 2
lines of symmetry
For an isosceles trapezoid , we have
one line of symmetry
10
Example 7: Real-World Connectio
Tell whether each object has rotational symmetry about a line and/or reflectional symmetry in a plane.
No
rotational
symmetry
The paddle has both rotational
and reflectional symmetry
The cup has reflectional
symmetry.
rotational and reflectional
symmetry
Example 8: 1)Does a regular octagon have
a) line symmetry?If so, how many lines of symmetry does it have? yes,8
b) rotational symmetry? If so, what is the angle of rotation?yes, 45 0
2) Which capital letters of the alphabet are rotational images of themselves? Draw each letter and give an angle
of rotation  360 0 Answer : H ,180 0 ; I ,180 0 ; O, any rotation ; X ,180 0 ; N ,180 0 ; S ,180 0 ; Z ,180 0


CONSTRUCTION OF REGULAR ANGLES ,
3 , 2 WITH JUST A RULER AND PENCILS
STEP1: In the coordinate plane, construct a unit circle of center O0,0 .(let assume here that
1 unit  8 cm ).This circle intercept the x-axis at E 1,0 and G 1,0 and the y-axis at D0,1 and U 0,1
STEP 2: Draw line x  half of the unit , y  half of the unit and line y  x
Step 3: line x  half of the unit cross the circle at A ( in the first quadrant) and B( in fourth quadrant).
3
Draw a line L parallel to the y-axis that intercept the circle at A;this line cut the y-axis at
and
2
m  EOA  60 0 Line y  half of the unit cross the circle at D( in first quadrant) and F( second quadrant)
3
and
2
m  EOD  30 0 Line y  x cross the circle at N( in first quadrant) and M( third quadrant). Draw line k that
Draw a line T parallel to the x-axis that intercept the circle at D;this line cut the y-axis at
cross the circle at N and is parallel to the y-axis. It cross the x-axis at
2
and m  EON  450 .
2
11
APPLICATION I
A “Stop” sign is a regular octagon. A regular octagon has eight congruent sides and eight congruent 135°
angles.
The figure shows an octagon with side length l in a coordinate plane so that one side falls along the x-axis
and one side falls along the y-axis. The coordinates of each vertex in terms of l is given by






 2  
 2 2
 
2   2 2  l 2
2 2 
A
l ,0  , B 0,
l  , C  0,
l  , D
, 1  2 l  , E 
l , 1  2 l  , F  1  2 l ,
l 
2
2
 2
  2  
  2
  2
 


l 2  2 2 
, H
G 1  2 l ,
l ,0  using a clockwise or counterclockwise rotation of 45°, 90°, 135°, or 180°
2   2


about its center, find the image of each point.We assume here l  2


12
APPLICATION II
A “Slow down” sign is a regular hexagon. A regular hexagon has six congruent sides and six congruent 120°
angles. The figure shows a hexagon with side length l in a coordinate plane so that one side falls along the
x-axis and one vertex falls on the y-axis. The coordinates of each vertex in terms of l is given by.
3  1
3  3 
1  
 3
 
l  , R l , l 3  , S  l , l 3  , T  2l ,
l  , U  l.0 
P l ,0  , Q 0,
2   2 
 
2   2  2
 2
using a clockwise or counterclockwise rotation of 45°, 90°, 135°, or 180° about its center, find the image of
each point. We assume here l  2
13
DILATION
Introduction:
you studied rigid transformations, in which the image and pre-image of a figure are congruent. In this lesson,
you will study a type of non-rigid transformation called a dilation, in which the image and preimage of a
figure are similar
Definition:
A dilation with center C and scale factor k is a transformation that maps every point P in the plane to
a point P' so that the following properties are true.
1. If P is not the center point C, then the image point P' lies on CP . The scale factor k is a positive
CP'
number such that k 
and k ≠ 1.
CP
2. If P is the center point C, then P = P'.
The dilation is a reduction if 0 < k < 1 and it is an enlargement if k > 1.
P' Q'
is equal to the scale factor of the dilation. In a coordinate plane,
PQ
dilations whose centers are the origin have the property that the image of P(x, y) is P' kx, ky .
Because PQR ~ P' Q' R' ,
Example 3: Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4).
1
Use the origin as the center and use a scale factor of . How does the perimeter of the preimage
2
compare to the perimeter of the image?
14
SOLUTION
Because the center of the dilation is the origin, you can find the image of each vertex by
multiplying its coordinates by the scale factor.
A(2, 2)  A' (1, 1)
(6, 2)  B' (3, 1)


C(6, 4)  C' (3, 2)
D(2, 4)  D' (1, 2)
From the graph, you can see that the preimage has a perimeter of 12 and the image has a perimeter of
6. A preimage and its image after a dilation are similar figures. Therefore, the ratio of the perimeters
of a preimage and its image is equal to the scale factor of the dilation.
Drawing a Dilation
Use the following steps to construct a dilation (k = 2) of a triangle using a straightedge
and a compass.
Draw PQR and choose
the center of the dilation C.
Use a straightedge to draw
lines from C through the
vertices of the triangle.
Connect the points P' , Q', and R'
Use the compass to locate P' on ray
CP so that CP' 2CP . Locate Q'
and R' in the same way
15
Example 4:
Draw a dilation of rectangle ABCD on a coordinate plane, with A(3, 1), B(3, 2.5),
C(5, 2.5), and D(5, 1). Use the origin as the center and use a scale factor of 2.
16
Guided Practice:
1) The vertices of ABC are A(-3,4) B(2,3), C(3,-2). The triangle is rotated 90 degrees counterclockwise.
0 - 1
Use the rotation matrix 
 [to find the coordinates of C'
1 0
2)The vertices of ABC are A(-3,4) B(2,3), C(3,-2). The triangle is rotated 30 degrees counterclockwise.
 3
1
- 

2
2
Use the rotation matrix 
[to find the coordinates of C"
1
3


2 
2
17
3) In a coordinate plane, sketch the quadrilateral whose vertices are A(2, 2), B(4, 1), C(5, 3), and D(3, 4).
Then, rotate ABCD 90° counterclockwise about the origin and name the coordinate
4) In a coordinate plane, sketch the quadrilateral whose vertices are A(2, 2), B(4, 1), C(5, 3), and D(3, 4).
Then, rotate ABCD 60° counterclockwise about the origin and name the coordinate.Use the rotation matrix.
18
Remark: In a translation a figure slides up or down, or left or right. No change in shape or size. The location
changes. All x and y coordinates of a translated figure change by adding or subtracting.
- In a reflection, a mirror image of the figure is formed across a line called a line of symmetry. No change in
size. The orientation of the shape changes. A reflection across the x -axis changes the sign of the y
coordinate. A reflection across the y-axis changes the sign of the x-coordinate.
- In a rotation, figure turns around a fixed point, such as the origin. No change in shape, but the orientation
and location change.
- Rules for rotating a figure about the origin
- 90 degree rotation:Rules for 90 degrees rotation- Switch the coordination of each point. Then change the
sign of the y coordinate. Flip order of x and y. Change signs according to what quadrant it's in.
Ex. A (2,1) to A’ ( 1,-2)
- Rotation of 90 degrees counter clockwise- Switch the coordinates of each point. Then change the sign of
the x- coordinate. Ex. B (3,1) to B’ (-1,3)
- Rotation of 180 degrees- Change the sign of both the x coordinate and the y coordinate.
Ex. C (4, 5) ,to C’(-4,-5)
- In dilation, a figure is enlarged or reduced proportionally. No change in shape, but unlike other
transformation, the size changes. In dilation all coordinates are divided or multiplied by the same number to
find the coordinates of the image.
GROUP WORK
• Students should complete the following example in pairs:
1)Rotation practice.
Rotate trapezoid ABC D clockwise 90 degrees and 270 degrees and give the vertex coordinates for each. Use
the following coordinate A ( 2, 1) , B (2,3), C ( 0,4), D (0,0). What is the new coordinate for prime one
ABCD at 90 degrees? Prime two ABCD at 270 degrees?
2) Reflection practice.
Triangle ABC has coordinate A (3,5), B (6,1), and C (1,2). Reflect triangle ABC across the x- axis and
determine the coordinate of triangle A’B’C’.
3)Dilation practice. Use a scale factor of 2 to dilate triangle ABC. Give the coordinates of the dilation.
Triangle ABC RULE 2 x.2 y  Triangle A’B’C’
, B2,4  B' .....,....
, C4,3  C' ....,....
A2.1  A' 4,2
1
4) Dilation practice :Use a scale factor to to dilate KLMN. Give the coordinates of the new vertices that
3
correspond to KLMN. K  6,6 , M 2,5 N 4,2 L0,2
5) Translation practice.
The vertices of a triangle are A ( 2,5), B(4,2), and C(0,2) . Give the vertices of triangle ABC translated to the
right 4 units and up 2. Locate points A,B,C and draw triangle ABC.
- On the x- axis, “right “ is a positive direction, so add 4 to each x coordinated.
- On the y- axis, “up” is a positive direction, so add 2 to each y-coordinate.
Locate points A,B,C and draw triangle ABC.
6) Translation practice.
The vertices of a triangle are A ( 2,5), B(4,2), and C(0,2) . Give the vertices of triangle ABC translated to the
left 4 units and down 2. Locate points A,B,C and draw triangle ABC.
- On the x- axis, “left “ is a negative direction, so subtract 4 to each x coordinated.
- On the y- axis, “down” is a negative direction, so subtract 2 to each y-coordinate.
19
Closure: Discussion:
• Describe how a figure’s coordinate change after the figure is translated to the left on a coordinate grid.
• Before drawing a dilation, how can you tell whether it will be an enlargement or a reduction?
• How can we summarize the rules that we’ve learnt today?
EXERCISE 1:
EXERCISE II
1)What is a center of rotation?
Use the diagram, in which ABC is mapped onto A' B' C' by a rotation of 90° about the origin.
2. Is the rotation clockwise or counterclockwise?
3. Does AB  A' B ' ? Explain.
4. Does AA'  BB" ?Explain.
5. If the rotation of ABC onto A' B' C' was obtained by a reflection of ABC in some
line k followed by a reflection in some line m, what would be the measure of the acute angle
between lines k and m? Explain.'
20
EXERCISE III :
The diagonals of the regular hexagon below form six equilateral triangles.
Use the diagram to complete the sentence.
6. A clockwise rotation of 60° about P maps R onto __________
7. A counterclockwise rotation of 60° about _________ maps
R onto Q.
8. A clockwise rotation of 120° about Q maps R onto _________.
9. A counterclockwise rotation of 180° about P maps V onto _________.
EXERCISE IV:
EXERCISE V: The vertices of octagon ABCDEFGH are
A2,1 , B1,2 , C  1,2 , D 2,1 , E 2,1 , F  1,2 , G1,2 , H 2,1
a) Draw ABCDEFGH on graph paper
b) Draw A' B' C' D' E' F' G' H' , the image of ABCDEFGH under D3 , on graph paper and write the
coordinates of its vertices.
c) Find HA, BC , DE , FG
d)Find H ' A' , B' C ' , D' E ' , F ' G '
e) if AB  CD  EF  GH  2 , find A' B' , C ' D' , E ' F ' , G ' H ' .
f)Are ABCDEFGH and A' B' C' D' E' F' G' H' similar polygons? Justify your answer.
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EXERCISE VI:
Let the vertices of ABC be A 2,3 , B 2,1 , and C3,1
a) find the area of ABC
b) Find the area of the image of ABC under D3
c) Find the area of the image of ABC under D4
d) Find the area of the image of ABC under D5
e) make a conjecture regarding how the area of a figure under a dilation Dk is related to the constant of
dilation k.
EXERCISE VII:
INDEPENDENT PRACTICE
1) A triangle has vertices at point A with coordinates (3, 1), B at (5, 3) and C at (1, 4). After the triangle is
rotated 180 degrees about the origin, what are the coordinates of its vertices?
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2) Given triangle ABC where A (–4, 1), B (– 2, 5) and C (0, 2),. Rotate the triangle 270°.
Then, sketch the image.
3) A figure has vertices X(-4, 1), Y(2, -2), and Z(-2, -4). Graph the figure and the image of the figure after a
rotation of 180 .
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4) Stella is designing a logo for her company. She wants to include two triangles, one of which is a rotation
of the other. She performed a rotation of 270 degrees to obtain a rotated triangle with coordinates of
3,1 , 4,4 and 1,4. Unfortunately, she spilled her coffee on the original triangle and only knows two of the
original coordinates,  4,1 and  4,4 . What is the third coordinate? Justify your answer.
5) The coordinates of Δ ABC are (-6,-2); (-6,-5); (-1,-5). What are the coordinates of the image A' B' C'
after a 180°
rotation around the origin?
6) The coordinates of Δ ABC are (-6,-2); (-6,-5); (-1,-5). What are the coordinates of the image
rotation around the origin? You may use the matrix
 3

 2
1

2
-1
2
3
2
A' B' C'
after a 30°






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7)Look at the parallelogram on the coordinate plane below.
a. Reflect ABCD across the y-axis. Name the reflected figure A'B' C'D' .
b. Translate A'B'C' D' four units downward. Name the translated figure A"B"C"D".
c. Write the coordinates of C ' and C".
d. If Px, y  , is on ABCD, what are the coordinates of the transformed point on
A”B"C"D" ? Explain how you determined your answer. Use words, symbols, or
both in your explanation.
8)Draw a rotation of 90° clockwise about the origin for the figure shown.
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9) Draw a rotation of 180° about the origin for the figure shown.
10) The original coordinates of a figure are (1,2); (-3,2); (8,0); (0,-8). After a size transformation, the new coordinates are
(5,10); (-15,10); (40,0); (0,-40). What was the factor of the size transformation?
11) Triangle XYZ is shown on the Cartesian plane. Draw a similar triangle with a size transformation of 2.
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12) Translate the triangle left 4 and down 5.
13)The coordinates of LMNO are  7.5 ; 0,5 ;  2,1 ;  5,1 .Find the reflection of L' M ' N ' O" of
LMNO over the y-axis then the 180 degree rotation of L' M ' N ' O" about the origin, call that image
L" M " N"O"
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14) Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g.,
graph paper, tracing paper, or geometry software
a. Reflect EDNA across the y-axis. Name the reflected figure A'B' C'D' .
b. Translate A'B'C' D' four units downward. Name the translated figure A"B"C"D".
c. Write the coordinates of C ' and C".
d. If Px, y  , is on ABCD, what are the coordinates of the transformed point on
A”B"C"D" ? Explain how you determined your answer. Use words, symbols, or
both in your explanation.
15)a. Reflect EDNA across the x-axis. Name the reflected figure E' D' N ' A' .
b. Translate A'B'C' D' two units downward and one unit right. Name the translated figure
E" D" N" A"
c. Rotate E" D" N" A" 90 degree clockwise about the origin . Name the rotation figure STLI
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16) Rotate STEA 90 0 counterclockwise about the origin. Name the image WILY . Rotate WILY 1800
across the origin. Name its image HRDG
17)Rotate LAUR 90 0 clockwise about the origin. Name the image ETIN . Rotate ETIN 270 0 across the
origin. Name its image HTSG
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18) Reflect LAUR across line x  4 . Name the reflected figure TCHE . Reflect TCHE across line
y   x Name the reflected figure GNTS . Write down all images coordinates
19) The vertices of ABC are A(-2, -5), B(1, -1), and C(-1, 3). Graph the triangle and the
image of ABC after a translation of 3 units right and 2 units up.
20) The vertices of the figure are M(-3, 4), N(-4, -2), O(3, 0), and P(2, 5). Graph the
figure and the image of the figure after a reflection over the y-axis.
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21) Draw quadrilateral ABCD with vertices A1,2 , B3,1 , C2,1 , and D 1,1 . Then find the
coordinates of the vertices of the image after a dilation having a scale factor of 3, and draw the image.
22) Draw PQR with vertices P(4, 4), Q(8, 0), and R6,2 . Then find the coordinates of the image
after a dilation having a scale factor of 0.5, and draw the image.
23)Find the scale factor of the dilation
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24) A triangle is dilated using a scale factor of 2, then its image is reflected in the y-axis.The figure is
the final image. Find the coordinates of the vertices of the original triangle, and draw the original
triangle.
CLOSURE
Tell whether a regular octagon has reflectional and/or rotational symmetry. eight lines of reflectional
symmetry and 45° rotational symmetry
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MULTIPLE CHOICE
Identify the letter of the choice that best completes the statement or answers the question.
1) Which type of isometry is the equivalent of two reflections across intersecting lines
a) rotation
b) reflection
c) glide reflection
d) none of these
Name the type of symmetry for the figure.
2)
a) rotational
3)
a) reflectional
b) reflectional
b) rotational
4)Which letter has rotational symmetry
a) H
b) Q
c) no symmetry
d) rotational and reflectional
c) reflectional and rotational
c)G
5) Which letter has at least one line of symmetry
a) U
b) S
d) no symmetry
d) A
c) Q
d) R
6) which figure, in general, has exactly two lines of symmetry
a) rectangle
b) pentagon
c) circle
d) square
7) which quadrilateral has rotational symmetry but not reflectional symmetry
a) parallelogram
b) nonisosceles trapezoid
c)kite
d) rhombus
8) what is the smallest angle through which you can rotate a regular hexagon onto itself?
a) 60 0
b) 30 0
c) 90 0
d) 120 0
9) Which figure does not have rotational symmetry
a)
b)
c)
d)
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10) Tell whether the tree-dimensional object has rotational symmetry about a line and/ or reflectional
symmetry in a plane
a) reflectional symmetry
b)rotational symmetry
symmetry and rotational symmetry
c) no symmetry
d) reflectional
11) How many lines of symmetry does the figure have?
a) 3
b) 2
c) 1
d) 0
12)Write a rule to describe the transformation that is a reflection in the y-axis
a) x, y    y, x
b) x, y   x, y 
c) x, y    x.  y 
d) x, y    x. y 
13) Write a rule to describe the transformation that is a reflection in the x-axis
a) x, y    y, x
b) x, y   x, y 
c) x, y    x.  y 
d) x, y    x. y 
14) Use an ordered pair to describe the translation that is 9 units to the right and 9 units up.
a)  9,9
b)  9,9
c) 9,9
d) 9,9
15) Use an ordered pair to describe the translation that is 1 units to the left and 10 units up.
a) 1,10
b)  1,10
c) 1,10
d)  1,10
16) Use an ordered pair to describe the translation that is 3 units to the left and 5 units down.
a)  3,5
b) 3,5
c) 3,5
d)  3,5
17)Describe in words the translation represented by the vector  3,8
a) 3 units to the right and 8 units down
b) 3 units to the right and 8 units up
c) 8 units to the left and 3 units up
d) 3 units to the left and 8 units up
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18) The vertices of a triangle are P2.8 , Q 8.  7 , and R5,6 . Name the vertices of the image
reflected in the x-axis.
a) P' 2,8, Q'  8,7, R' 5,6
b) P'  2,8, Q' 8,7, R'  5,6
c) P'  2,8, Q' 8,7, R'  5,6
d) P' 2,8, Q'  8,7, R' 5,6
19) The vertices of a triangle are P 4.  6 , Q 3.3 , and R4,5 . Name the vertices of the image
reflected in the line y  x .
a) P'  6,4, Q' 3,3, R'  5,4
b) P' 6,4, Q'  3,3, R' 5,4
c) P' 6,4, Q'  3,3, R' 5,4
d) P'  6,4, Q' 3,3, R'  5,4
20) The vertices of a triangle are P7,1 , Q6,5 , and R 4,1 . Name the vertices of the image
reflected in the y-axis.
a) P'  7,1, Q'  6,5, R' 4,1
b) P' 7,1, Q' 6,5, R'  4,1
c) P'  7,1, Q'  6,5, R' 4,1
d) P' 7,1, Q' 6,5, R'  4,1
21) The dashed triangle is a dilation image of the solid triangle. What is the scale factor?
a)
1
2
b)
1
4
c)
2
3
d) 2
22) Multiple choice: Let P2,4 be a point on a figure, and let P ' be the corresponding point on the
image. The figure is dilated by a scale factor of 4. What are the coordinates of P ' ?
1 
a)  2,0
b)  ,1
c) 6,8
d) 8,16
2 
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23) Which translation from solid-lined figure to dashed-line figure is given by the vector  3,3
a)
b)
d)
c)
24) Quadrilateral WXYZ is located at W(3, 6), X(5, -10), Y(-2, -4), Z(-4, 8). A rotation of the quadrilateral is
located at W’(-6, 3), X’(10, 5), Y’(4, -2), Z’(-8, -4). How is the quadrilateral transformed?
a) Quadrilateral WXYZ is rotated 90º counterclockwise about the origin
b) Quadrilateral WXYZ is rotated 90º clockwise about the origin
c) Quadrilateral WXYZ is rotated 180º about the origin
d) Quadrilateral WXYZ is rotated 45º about the origin
hint: Quadrilateral WXYZ is rotated 90º counterclockwise about the origin. Just by looking at the
coordinates of the points (and how they y coord is now the x coord and vice versa). To confirm this, you can
graph the points before and after the rotation and noticed the 90 degree counter-clockwise rotation.
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25) Name the translation image of ABC after a refection over a line t and then a reflection over line
r.
a) UVW
b) LMN
c) DEF
d) XYZ
26) In the diagram, quadrilateral A' B' C' D' is the image of quadrilateral ABCD after a dilation. What
is the scale factor
a)
1
4
b)
1
2
c) 2
d) 3
27) Which transformation represents a dilation?
a) 8,4   4,8
b) 8,4   8,4
c) 8,4  11,7
28) What is the image of (1,-6) for a 90º counterclockwise rotation about the origin?
a) 6,1
b)  1,6
c)  6,1
d) 8,4  4,2
d)  1,6
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29) The figure is the front view of one of the dolls in a set of nesting dolls. Draw the outline of the
figure. Then, on the same coordinate plane, draw the images of the outline afterdilations having the
1 1
following scale factors: , 1 ,2
2 2
30) The accompanying diagram shows the starting position of the spinner on a board game.
How does this spinner appear after a 270 degree counterclockwise rotation about point P?
a)
b)
c)
d)
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The hexagon GIKMPR and FJN are regular. The dashed line segments form 30 0 angles.
31) Find the image of ON after a rotation of 180 0 about point O
a) OL
b) OF
c) OH
32) Find the angle of rotation about O that maps QR to HI
a) 210 0
b) 270 0
c) 240 0
33) Find the angle of rotation about O that maps Q to N
a) 90 0
b) 30 0
c) 60 0
d) OJ
d) 120 0
d) 300 0
34) Find the image of OQ after a rotation of 240 0 about point O
a) OF
b) OH
c) OJ
d) OL
35) If point (5, 2) is rotated counterclockwise 90 degree about the origin, its image will be point
a) 2,5
b)  2,5
c)  5,2
d) 2,5
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36) find the degree of rotation about the spinner center that maps label h to label B
a) 216 0
b) 1440
c) 252 0
d) 180 0
37) If the figure has rotational symmetry, find the angle of rotation about the center that results in an
image that matches the original figure.
a) 120 0
b) 90 0
c) no symmetry
d) 210 0
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38)
38) State the segment that represents a 90 0 clockwise rotation of AB about P.
a) BC
c) AH
b) CD
d) GF
39) State the segment that represents a 900 counterclockwise rotation of CE about E.
a) AC
c) GE
b) AG
d) BF
40)State the triangle that represents a 180 0 rotaton of triangle KEF about P.
a) MHA
b) KDE
c) JCD
d) MAB
Hint
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TEST:
Triangle ABC has vertices A(-2, 3) , B(2, 4) and C(2,0).
1. Find the coordinates of ABC after translating right 3 and down 1.Name the image STE
2. Find the coordinates of STE after reflecting it over the x-axis. Name this image LUV
3. Find the coordinates of LUV after reflecting it over y = - x .Name the image ZMN
4. Find the coordinates of ZMN after 270 degree rotation about the origin. Name this image RPQ
5.
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BCR: On the accompanying set of axes, draw ABC , whose coordinates are A(-7,9), B(-2,8), and C(-3,4).
Then draw, label, and state the coordinates of A' B' C' , the image of ABC after the transformation that
maps x, y  to  x, y  . Based on your diagram, identify the type of transformation that was performed.
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EXERCISES FROM TEXTBOOK Dilation:pages 676 - 679 #1-9,15-40,43-45 ,47-48, 64, 67-71
Page 683 #36-40
rotation: p 680 #1-3 , page 682 # 18-22 , p 651 #35-39
Reflection P 682 # 8-14 Page 684 #1-14 ; page 685 # 7
Pages 701 # 1-8, 9-17, 22-29
http://www.regentsprep.org/Regents/math/geometry/MultipleChoiceReviewG/Transformations.htm
http://www.nexuslearning.net/books/ML-Geometry/Chapter8/ML%20Geometry%208-7%20Dilations.pdf
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