GEOMETRY TRANSFORMATION
objective: Students will be able to
Students will be able to identify and compare the three congruence transformations,
apply the three congruence transformations to coordinates of the vertices of figures,
identify and apply dilations, and apply transformations to real-world situations.
- represent /draw and interpret the results of transformations and successive
transformations on figures in the coordinate plane.
•
translations
•
reflections
•
rotations (90°, 180°, clockwise and counterclockwise about the origin)
•
dilations (scale factor)
- identify locations, apply transformations, and describe relationships using coordinate geometry.
Compare transformations that preserve distance and angles to those that do not.
Solve problems involving transformations in order to solve real-world problem.
Ability to make connections between function transformations (F.BF.3) and geometric
transformations
transformations preserve the shape of a figure
Ability to use appropriate vocabulary to describe the rotations and reflections
determine and then describe what happens to the
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figure as it is rotated (such as axis of symmetry, congruent angles or sides….
Ability to interpret and perform a given sequence of transformations and draw the result
describe the sequence of transformations that
will carry a given figure onto another
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)
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
2
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,
Manipulation of
rotation
geometer's sketchpad
and what does it enable us to
geometric figures can
reflection
graph paper
understand?
be a useful tool in real
dilation
journal
Why is it important to be able
world situations
translation
straight edge
to
glide reflection
move a shape?
image
Worksheets,
What are the similarities and
preimage or original
protractor, ruler, patty
differences between the
rotation reflection
paper, Mira™
images and pre-images
translation translation
Optional – Dynamic
generated by translations?
vector
geometry softwar
What is the relationship
direction distance
between the coordinates of the
angle of rotation
vertices of a figure and the
center of rotation
coordinates of the vertices of
line of reflection rigid
the figure’s image generated
transformation
by translations?
How can translations be
applied to real-world
situations?
3
2
WARM UP: Use the matrices below to answer the question A  1

5
3
 1

0 , B  2
3
4 
0
1
4 
Which matrix represents the expression 2 B  A ?
 4
a)  2

 10
-6 
0 
- 8
 4
b) 3

1
 6
c) 2

 4
- 3
2 
4 
- 6
2 
0 
5
d) 0

7
6
- 1
4 
Definition:
Reflection over a line k (notation rk) is a transformation
in which each point of the original figure (pre-image) has
an image that is the same distance from the line of reflection
as the original point but is on the opposite side of the line.
Remember that a reflection is a flip. Under a reflection, the
figure does not change size. rk ABC   A' B' C '
The line of reflection is the perpendicular bisector of the
segment joining every point and its image.
A line reflection creates a figure that is congruent to the original figure and is called
an isometry (a transformation that preserves length). Since naming (lettering) the figure in a
reflection requires changing the order of the letters (such as from clockwise to
counterclockwise), a reflection is more specifically called a non-direct or opposite isometry.
Properties preserved (invariant) under a line reflection:
1. distance (lengths of segments are the same)
2. angle measures (remain the same)
3. parallelism (parallel lines remain parallel)
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4. colinearity (points stay on the same lines)
5. midpoint (midpoints remain the same in each figure)
6. orientation (lettering order NOT preserved. Order is reversed.)
Reflecting over the x-axis: (the x-axis as the line of reflection)
When you reflect a point across the x-axis, the x-coordinate remains the same,but the y-coordinate is
transformed into its opposite.
The reflection of point x, y  across the x-axis is point x, y  . M x, y   rxaxis x, y   M ' x, y 
Hint: If you forget the rules for reflections when graphing, simply fold your graph paper along the line of
reflection (in this example the x-axis) to see where your new figure will be located. Or you can measure how
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far your points are away from the line of reflection to locate your new image. Such processes will allow you
to see what is happening to the coordinates and help you remember the rule.
Reflecting over
y  k : (parallel to x-axis)
When you reflect a point across y  k , the x-coordinate remains the same,but the y-coordinate is
transformed into 2k-y.
The reflection of point x, y  across the x-axis is point x,2k  y  . M x, y   rx axis x, y   M ' x,2k  y 
Reflecting over the y-axis: (the y-axis as the line of reflection)
When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is
transformed into its opposite. The reflection of point x, y  across the y-axis is point  x, y  .
M x, y   ry axis x, y   M '  x, y 
Reflecting over the
x  k : (parallel to the y-axis )
When you reflect a point across x  k , the y-coordinate remains the same, but the x-coordinate is
transformed into 2k  y . The reflection of point x, y  across the y-axis is point
2k  x, y  . M x, y   ry axis x, y   M ' 2k  x, y 
Reflecting thru the origin also call 180 degree rotation
6
When you reflect a point across the origin, the y-coordinate is transformed into its opposite , and the xcoordinate is transformed into its opposite. basically, just change signs. The reflection of point x, y  across
the origin is point  x, y  . M x, y   rorigin x, y   M '  x, y 
Reflecting thru a different point
The reflection of point x, y  across a point h, k  is point 2h  x,2k  y .
M x, y   rh.k  x, y   M ' 2h  x,2k  y 
Reflecting over the line y = x or y = -x:
(the lines y = x or y = -x as the lines of reflection)
When you reflect a point across the line y = x, the x-coordinate and the y-coordinate change places. When
you reflect a point across the line y = -x, the x-coordinate and the y-coordinate change places and are negated
(the signs are changed).
M x, y   ry  x x, y   M '  y, x 
The reflection of the point x, y  across the line y  x is the
point  y, x .
The reflection of the point x, y  across the line y   x is the
point  y, x .
M x, y   ry  x x, y   M '  y, x 
Reflecting over any line:
Each point of a reflected image is the same distance from the line of reflection as the
corresponding point of the original figure. In other words, the line of reflection lies directly in
the middle between the figure and its image -- it is the perpendicular bisector of the segment
joining any point to its image. Keep this idea in mind when working with lines of reflections
that are neither the x-axis nor the y-axis.
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Each point of the original figure and its image are the same distance away from the line of reflection (which
can be easily counted in this diagram since the line of reflection is vertical).
90 degree rotation
Flip order of x and y. Change signs according to what quadrant it's in.
ROTATION ABOUT A POINT WHICH IS NOT THE ORIGIN
Rotation is the transformation along circular paths. Rotation angle  dtermines the amount of rotation for
each vertex.
- sin  
cos 
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 maps it onto a point M ' x' , y' such that
- sin    x 
 x'  cos 
which is the rotation of the matrix about the origin through an angle  .
 y '  sin 
cos   y 
  
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: 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
1
, Sin 30 0   . Rotation matrix is given by
2
2
 3
1
- 

- sin 30
2
 2

cos30   1
3


2 
2
8
Substituting the above values
 3
 x'   2
 y '  
  1

2
1
-  10
2     4.5 3 

  
7 
3   3  

2 
Therefore, the image point is given by M ' 4.5 3 ,7 
INDEPENDENT PRACTICE
Graph the image of the figure using the transformation given
1) reflection across the x-axis
2) reflection across y = 3
reflection across y = 1
reflection across the x-axis
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reflection across the x-axis
reflection across y = −2
T(2, 2), C(2, 5), Z(5, 4), F(5, 0)
H(−1, −5), M(−1, −4), B(1, −2), C(3, −3)
Write a rule to describe each transformation
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Draw the image according to the rule and identify the type of transformation
x, y   x, y
x, y   x  2, y  5
x, y  2  x, y 
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x, y    x, y 
x, y   x, y 
x, y    1 x, 1 y 
2
2 
Complete the ordered pair rule that transforms each triangle to its image. Identify the transformation.
Find all missing coordinates.
12
BCR: An air traffic control system at Little Rock National (LIT) airport, located at (2, 3) on the grid below,
uses a radar system that sends out signals to determine the locations of airplanes. This system can detect
planes within a circular region having a radius of 35 miles from LIT. Each grid unit represents 5 miles.
An airplane is heading directly toward LIT from the location represented by coordinates  4,7 on the grid.
1. Can the plane be detected by the radar? Support your answer with mathematical evidence.
2. The air traffic controller instructs the pilot to begin circling the airport halfway between the airport
and her current location. What will be the coordinates of the plane’s location when the pilot begins
to circle the airport? Show your work or explain your answer.
MULTIPLE CHOICES
1) RST is shown below
13
If RST is rotated clockwise about the origin, what will be the coordinates of the image of point T?
A) 6,2
b)  6,2


0
 x'  cos  180

 y ' 
0
  sin  180




c) 6,2


- sin - 180 0  6
 x'   1




 y '  0
cos - 180 0  2
  
*d)  6,2
0  6 
 x'   6 
  



- 1 2
 y '  2
2) Use the graph below to answer question
14
Which graph shows a reflection across the x-axis of the image above?
a)
b)
3) The figure below is rotated 90 0 clockwise about the origin, then reflected across the y-axis.
15
What is the final figure?
4) Josh is designing a cover for a paperback book. He is going to use the graphic shown above. He plans to
reflect the graphic over the y-axis. What will be the coordinates of the reflection of point A?
16
a)  5,8
*b) 5,8
c) 5,8
d) 8,5
5) Roberto is a computer graphics designer and is working on an ad for the local coffee shop. The figure
above shows a coffee mug in two different positions. Which describes the transformation of the coffee mug
in position I to the image in position II?
A. a reflection over a horizontal line and a translation down
* B. translation down and a reflection over a vertical line
C. 180° rotation
D. translation to the right and a reflection over a vertical line
6) The figure graphed below is rotated 90 0 clockwise about the origin and translated up 1 unit.
17
Which is the resulting image?
a)
b)
c)
d)
7) Which image will result from the figure below being rotated 90q clockwise about the origin and then
18
reflected across the y-axis?
a)
b)
8) Polygon STUVW is shown below.
19
After polygon STUVW is reflected across the y-axis, what are the coordinates of S′, the image of point S
after the transformation?
A. (−5, −2)
B. (−5, 2)
* C. (5, −2)
D. (5, 2)
9) The figure below is translated 3 units to the right, then 5 units down, and finally reflected over the x-axis.
What are the coordinates of the image of point X after the transformations?
A. (−2, −2)
B. (−2, 2)
C. (2, −2)
D. (2, 2)
10) RST is shown below.
20
If RST is rotated 180 0 clockwise about the origin, what will be the coordinates of the
image of point T?
a) 6,2
b)  6,2
c) 6,2
d)  6,2
11) If the parallelogram below were translated 3 units left and 6 units down, what would be
the coordinates of the new image W′X′Y′Z′?
A. W′(–2, –1), X′(0, 3), Y′(5, 3), Z′(3, –1)
C. W′(4, –1), X′(6, 3), Y′(11, 3), Z′(9, –1)
B. W′(–1, –2), X′(3, 0), Y′(3, 5), Z′(–1, 3)
D. W′(7, 8), X′(9, 12), Y′(14, 12), Z′(12, 8)
12) Triangle JKL is translated 4 units left and 5 units up. What are the coordinates of the
21
image of point J?
A. (2, 6)
B. (3, –3)
* C. (– 6, 6)
D. (–2, 6)
13) What would the figure below look like if it were reflected over the x-axis?
a)
b)
c)
d)
14) Triangle QRS is shown in the graph below.
22
Which of the following graphs shows ∆QRS rotated 90 degrees counterclockwise about the origin?
a)
b)
c)
d)
15) Segment JK JK is reflected across the y-axis to form J ' K ' . What are the coordinates of J′ and K′
23
A. J′(– 4, –5), K′(–2, 1)
B. J′(5, – 4), K′(–1, 2)
C. J′(4, 5), K′(–2, 1)
* D. J′(5, 4), K′(1, –2)
16) The arrow above represents the needle on a compass. The needle is rotated 180° in the clockwise
direction. What are the coordinates of point A after the rotation?
A. (–8, –6)
B. (–8, 6)
* C. (–6, –8)
D. (6, –8)
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17) Triangle PQR has vertices of P(–2, –1), Q(1, 6), and R(3, –2). What are the coordinates of the vertices of
the image of ∆PQR if the figure is translated 4 units right and 3 units up?
A. P′(2, 2), Q′(5, 9), R′(7, 1)
B. P′(1, 3), Q′(4, 10), R′(0, 2)
C. P′(– 6, 2), Q′(–3, 9), R′(–1, 1)
D. P′(– 8, –3), Q′(4, 18), R′(12, – 6)
18) The point of the heart (H) has a coordinate of (–5, –7) as shown above. The heart is reflected over the yaxis and then reflected over the x-axis. After both reflections, what are the coordinates of the point H?
A. (–5, –7)
B. (–5, 7)
C. (5, –7)
* D. (5, 7)
25
19) The polygon above is the mapping of a school building. What translation rule moves point A
to the point (0, 0)?
A. x, y   x  4, y  2 B. x, y   x  2, y  4 C. x, y   x  4, y  2 D. x, y   x  0, y  0
20) Which transformation describes the change from Figure M to Figure N?
A. dilation
* B. reflection
C. rotation
D. translation
26
21) Sacha planned a fabric design by reflecting the triangle shown above over the x-axis. Which
list of coordinates represents the vertices of the triangle reflected over the x-axis?
A. (–2, –3), (–4, –6), (–8, 1)
C. (2, –3), (4, –6), (8, –1)
B. (–2, 3) , (–4, 6), (–8, 1)
D. (3, 2), (6, 4), (1, 8)
22) Which would move Flag A to Flag B in the graph below?
*A. clockwise rotation of 180°
B. reflection over x-axis, then clockwise rotation of 90°
C. translation 8 units left and 7 units down
D. reflection over y-axis, then translation 8 units left and 7 units down
27
23) To plan a scene in an animated movie, Roger rotates the below figure around point P by 90° in a
clockwise direction. Which drawing shows the pre-image and the final image?
a)
b)
c)
d)
28
24) A quilt design is formed by translating a polygon across the coordinate plane as shown in the figure
below. Which is a translational rule that will translate point A to point B?
A. x, y   x  0, y  8
B. x, y   x  4, y  2 C. x, y   x  2, y  3 * D. x, y   x  4, y  2
25) After this translation x, y   x  3, y  2 of point T, what are the coordinates of the new point?
A. (–1, 3)
B. (0, 4)
C. (3, –2)
D. (5, –1)
29
26) Janelle and Franz are playing a game. The rule of the game is each playing piece must go through
one transformation. Janelle reflects piece X over line 3. Where will piece X land?
A. on piece A
* B. on piece B
C. on piece C
D. on piece D
27) Which graph represents the figure below reflected across the y-axis?
a)
b)
c)
d)
30
28) The coordinates of C are (--4, 1). Which translation moves ABC to A' B' C' ?
A. translate 8 units left, 4 units up
C. translate 3 units right, 8 units down
B. translate 3 units left, 8 units up
* D. translate 8 units right, 4 units down
29) The point on the grid is reflected across the y-axis. What are its new coordinates?
A. (--5, 2)
B. (--2, 5)
C. (2, --5)
D. (2, 5)
30) After translating the point (x, y) four units to the right, what are its new coordinates?
A. x, y  4
B. x, y  4
C. x  4, y 
D. x  4, y 
31
31) Which figure represents a rotation of figure 1?
A. figure 2
* B. figure 3
C. figure 4
D. figure 5
32) What are the coordinates of PQRS after a translation of 6 units to the right and 6 units up?
a) P0,3 , Q3,3 , R3,3, S 0,3
c) P0,3 , Q3,3 , R3,0 , S 0,0
b) P 6,3 , Q 3,3 , R 3,0 , S  6,0
d) P 3,0 , Q0,0 , R0,3 , S  3,3
32
33)Triangle ABC is reflected over the x-axis. What will be the coordinates of A', B', and C'?
a) A'  1,1 , B' 1,3 , C ' 2,1
A' 2,1 , B' 1,3 , C'  2,1
b) A' 2,1 , B' 1,3 , C '  2,1 c)
d) A' 1,3 , B' 2,1 , C'  2,1
34) Which shadow shows a reflection of the corresponding figure?
a)
b)
*c)
d)
33
35)Which of the following figures represents a 270 0 clockwise rotation of the flag about point A?
b)
a)
c)
d)
34
36) Which is a reflection of the below figure across the x-axis?
a)
c)
b)
d)
35
37)
B'?
a) b) *c) d)
AB is translated 3 units left and 1 unit down to form A' B' . What are the coordinates of A' and
a) A'  4,1 , B' 2,0
b) A'  4,2 , B' 2,0
c) A' 1,2 , B' 0,2
d) A' 2,2 , B' 0,2
38) If point P has coordinates  3,7 as shown on the grid below, what are the coordinates of the point
P', the reflection of P across the x-axis?
a) 3,7 
b)  3,7
c) 3,7
d)  3,7
36
39) Which of the choices is a reflection over the y-axis of the triangle shown below ?
b)
a)
c)
d)
The image of the point (4,-3) under a reflection across the x-axis is
(-4,-3).
a) true
b) false
40)
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41) The image of the point (-5,4) under a reflection across the y-axis is
(5,4).
a) true
b) false
42) The image of the point (-1, 8) under a reflection across the line y =
x is (8,-1).
a) true
b) false
43) This graph illustrates a reflection over the x-axis.
a) true
b) false
44) This graph illustrates a reflection over the x-axis.
a) true
b) false
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45) BCR:
Triangle ABC has coordinates A(-3,3), B(3,2) and C(-1,-4). If the
triangle is reflected over they-axis, what are the coordinates of image
triangle A'B'C'?
46)
The line of reflection between these two triangles is y = -1.
a) true
b) false
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47) BCR: a.
Which point is a reflection of point A over the x-axis?
b. Which point is a reflection of point A over the y-axis?
c. Which point is a reflection of point A over the line y = x?
This L-shaped figure is reflected over the y-axis. The image is
then reflected over the x-axis. What are the coordinates of the vertices
of the final image?
48)BCR
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Which of the following descriptions (pertaining to the graph at the
right) is true?
49)
a) A" B"C" is a translation of ABC
c) ) A" B"C" is a reflection in the origin of ABC
b) A" B"C" is a glide reflection of ABC
d) A" B"C" is a dilation scale factor 2 of ABC
Which of the following transformations is illustrated by the graph at
the right?
50)
dilation
b) reflection in y = x
d) reflection in the origin
a)
c) translation
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51) Which of the following descriptions does NOT pertain to the
transformation shown at the right?
a) translation of vector  7,3
b) x, y   x  7, y  3
c) 7 units down and 3 units left.
An isometry is a transformation of the plane that preserves length.
a) true
b) false
53) If a reflection in the line y = -x occurs, then the rule for this
reflection is:
52)
a) x, y   x, y 
b) x, y    x, y 
c) x, y    y, x 
d) x, y    y, x
54) BCR: Angle
ABC has been reflected in the x-axis to create
angle A'B'C'. Prove that angle measure is preserved under a reflection.
Answer
Let's see if this is true using slopes.
Since these slopes are negative reciprocals, these segments are perpendicular, meaningm<ABC =
90º.
,
. Since these slopes are
negative reciprocals, these segments are perpendicular, meaningm<A'B'C' = 90º. Angle measure
is preserved.
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