9.1 Geometry Transformations
Transformations
Transformations are functions that take points in the plane as inputs and give other points as outputs. Some
transformations preserve distance and angles and some do not. You will learn how to predict whether or not a
transformation will maintain the congruence between the pre-image and image – i.e., the input object is
congruent to the output object.
Note: You may want to use geometry software such as Geometer’s Sketchpad or Geogebra.
Transformation Terminology
pre-image: original figure before
transformation; input into the transformation
function
image: figure after transformation; output
from the transformation function
Rigid Motion Transformations
Rigid motion transformation are transformations that preserve congruence.
Reflection (flip)
 The reflection of an object is called its image. If the original object (the pre-image) was labeled with
letters, such as polygon ABCD, the image may be labeled with the same letters followed by a prime
symbol, A'B'C'D'.
 The line where a mirror may be placed is called the line of reflection. The distance from a point to the
line of reflection is the same as the distance from the point's image to the line of reflection.
 A reflection can be thought of as folding and "flipping" an object over the line of reflection.
Notice that the pre-image B and the image B' define a
line that is perpendicular to the line of reflection. The
point of intersection is also the midpoint of BB ' .
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Translation (slide)
 A translation "slides" an object a fixed distance in a specified direction. The original object and its
translation have the same shape and size, and they face in the same direction. The word translate in
Latin means carried across.
 Translations move a point a specified distance along a line parallel to a specified line.
 If the pre-image (input) was polygon ABCDE, then the image after a translation has taken place would
be polygon A'B'C'D'E'.
Think of polygon ABCDE as sliding two
inches to the right and one inch down. Its
new position is labeled A'B'C'D'E'.
Notice that the lines m, CC ' and EE ' are
parallel.
Rotation
A rotation turns an object around a point. Rotations can occur in either a clockwise (to the right) or
counterclockwise (to the left) direction.
 A positive angle of rotation turns the figure counterclockwise, a negative angle of rotation turns the
figure clockwise.
 Rotations move objects along a circular arc with a specified center through a specified angle.
 Notice that a rotation does not change the size of the figure.
This rotation of the pre-image polygon
ABCDE is 90° counterclockwise. The output is
the image A'B'C'D'E'.
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Non-Rigid Motion Transformations
Non-rigid transformations are transformations that do not preserve congruence.
Horizontal Stretch
A horizontal stretch distorts a plane figure horizontally.
 The size and shape of the figure is changed so the image and pre-image are no longer congruent.
Vertical Stretch
A vertical stretch distorts a plane figure vertically.
 The size and shape of the figure is changed so the image and pre-image are no longer congruent
Reflection in the Coordinate Plane
Note: A reflection in the origin is the same thing as a 180° rotation about the origin.
Line of
reflection
Pre-image to
image
How to find
coordinates
x-axis
y-axis
origin
yx
 a, b    a, b 
 a, b    a, b 
 a, b    a, b 
 a, b    b, a 
Multiply the ycoordinate by -1.
Multiply the xcoordinate by -1.
Multiply both
coordinates by -1.
Interchange the
x- and y-coordinates.
Example
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Example (transformation function)
Here is an example of a transformation function. The original triangle that is graphed is the input or pre-image.
The points for the vertices of the triangle are used as input into the function and the result is a new triangle (or
output).
Graph the triangle and its image under the given translation.
∆EFG with vertices E(0, -4), F(-4, -4), and G(0, 2) under the translation  x, y    x  2, y  1 .
E  0  2, 4  1  E '  2, 5 
F  4  2, 4  1  F '  2, 5 
G  0  2, 2  1  G '  2,1
NOTE: A composition of reflections across two parallel lines (or across any even number of parallel lines) is
equivalent to a translation. A composition of reflections across three parallel lines (or across any odd number of
parallel lines) is equivalent to a single reflection. A composition of reflections over intersecting lines is
equivalent to a rotation.
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9.1—Transform Points and Shapes in the Coordinate Plane
Use this blank page to compile the most important things you want to remember for cycle 9.1:
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9.1a (refine)— Transform 2D
1. Describe the transformations (rotations, reflections) using the L-Shape in BOX 1 as the starting point.
Estimate the degree of rotation. If there is more than one way to describe the transformation, note the
different ways.
Starting point
e.
a.
b
c.
d.
f.
g.
h.
i.
2. Transform the L-Shape show at right below.
Where will the L-Shape be if it is:
a. reflected over the x-axis?
b. reflected over the y-axis?
c. reflected over the line y  x ?
d. reflected over the line y   x ?
e. reflected over the line x  2 ?
f. rotated 180  around the origin?
g. rotated 90  clockwise around the origin?
h. rotated 90  counter-clockwise around (3, 0) ?
i. translated +3 horizontally, -4 vertically?
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9.1b (build)—Intro to Transformations—Begin to Develop Rules
1. On graph paper, create a small coordinate grid. Locate and mark the starting point (2, 6). Now locate, mark,
and label the coordinates of the point (2, 6) after it is:
a. Reflected over the x-axis.
b. Reflected over the y-axis.
c. Rotated 180° about the origin.
d. Reflected over the line y  x .
e. Reflected over the line y   x .
f. Rotated 90° clockwise about the origin.
g. Rotated 90° counterclockwise about the origin.
h. Rotated 90° clockwise about the point (-2, 1).
i. Translated 3 units vertically and -2 units horizontally.
2. Create a second coordinate grid. Select any point and make it the point (x, y) on the grid. Now locate and
mark the point (x, y) after it is:
a. Reflected over the x-axis.
b. Reflected over the y-axis.
c. Rotated through 180° about the origin.
d. Reflected over the line y  x .
e. Reflected over the line y   x .
f. Rotated 90°clockwise about the origin.
g. Rotated 90°counterclockwise about the origin.
h. Rotated 90° clockwise about the point (3, -1).
i. Translated -4 units vertically and 3 units horizontally.
3. Write the transformation rules for each transformation below.
a. Reflected over the x-axis.
b. Reflected over the y-axis.
c. Rotated through 180° about the origin.
d. Reflected over the line y  x .
e. Reflected over the line y   x .
f. Rotated 90° clockwise about the origin.
g. Rotated 90° counterclockwise about the origin.
h. Rotated 90° clockwise about the point (3, -1).
i. Translated -4 units vertically and 3 units horizontally.
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9.1c (refine)—Transform Points
Using the rules you developed, write the new coordinates of the given point when the given transformation is
performed.
1. Point (5, 3)
a. Reflected over the x-axis
b. Rotated 180○ about the origin
c. Translated up 2 units
2. Point (8, -2)
a. Reflected over the y-axis
b. Rotated 90○ clockwise about the origin
c. Translated right 10 units
3. Point (6, -10)
a. Reflected over the line y = x
b. Rotated 90○ counterclockwise about the origin
c. Translated down 6 units
4. Point (-4, 8)
a. Reflected over the line y = -x
b. Rotated 270○ clockwise about the origin
c. Translated left 2 units
5. Point (4, -3)
a. Reflected over the x-axis
b. Rotated 90○ clockwise about the origin
c. Translated up 7 units and right 3 units
6. Point (1, 0)
a. Reflected over the line y = x
b. Rotated 270○ clockwise about the origin
c. Translated down 1 unit and left 1 unit
7. Point (1, 8)
a. Reflected over the y-axis
b. Rotated 180○ about the origin
c. Translated left 7 units and up 2 units
8. Point (9, -5)
a. Reflected over the line y = -x
b. Rotated 90○ counterclockwise about the origin
c. Translated right 6 units and down 3 units
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9.1d (refine)—Translations in the Coordinate Plane
Use the translation  x, y    x  5, y  9  for questions 1–7.
1. What are the coordinates of the image of
2. What are the coordinates of the image of B  4,8 ?
A  6,3 ?
3. What are the coordinates of the image of
4. What are the coordinates of the pre-image of
C  5, 3 ?
D ' 12, 7  ?
5.
What are the coordinates of the image of A ' ?
7.
Plot
6.
What are the coordinates of the image of
?
from the questions above. What do you notice?
The vertices of ABC are A  6, 7  , B  3, 10  and C  5, 2  . Find the vertices of A ' B ' C ' , given the
translation rules below.
8.
 x, y    x  2, y  7 
9.
 x, y    x  11, y  4
10.
 x, y    x, y  3
11.
 x, y    x  5, y  8
12.
 x, y    x  1, y 
13.
 x, y    x  3, y  10
In questions 14–17, A ' B ' C ' is the image of ABC . Write the translation rule.
14.
15.
16.
17.
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18. If A ' B ' C ' were the pre-image and ABC were the image, write the translation rule for #14.
19. If A ' B ' C ' were the pre-image and ABC were the image, write the translation rule for #15.
20. Find the translation rule that would move A to A '  0, 0  , for #16.
21. The coordinates of DEF are D  4, 2  , E  7, 4  and F  5,3 . Translate DEF to the right 5 units and
up 11 units. Write the translation rule.
22. The coordinates of quadrilateral QUAD are Q  6,1 , U  3,7  , A  4, 2  and D 1, 8 . Translate
QUAD to the left 3 units and down 7 units. Write the translation rule.
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9.1e (refine)—Reflections in the Coordinate Plane
Reflect each shape over the given line.
1. y-axis
2. x-axis
3. y  3
4. x  1
5. x-axis
6. y-axis
7. y  x
8. y   x
9. x  2
10. y  4
11. y   x
12. y  x
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Find the line of reflection between the pre-image and the image.
13.
14.
15.
pre-image
pre-image
pre-image
The vertices of ABC are A(-5, 1), B(-3, 6), and C(2, 3). Use this information to answer questions 16–19.
16. Plot ABC on the coordinate plane.
17. Reflect ABC over y  1. Find the coordinates of
A ' B ' C ' .
18. Reflect A ' B ' C ' over y  4 . Find the coordinates of
A '' B '' C '' .
19. What one transformation would be the same as this double
reflection?
The vertices of DEF are D(6, -2), E(8, -4), and F(3, -7). Use this information to answer questions 20-23.
20. Plot DEF on the coordinate plane.
21. Reflect DEF over x  2 . Find the coordinates of
D ' E ' F '.
22. Reflect D ' E ' F ' over x  4 . Find the coordinates of
D '' E '' F '' .
23. What one transformation would be the same as this double
reflection?
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9.1f (refine)—Rotations in the Coordinate Plane
1. If you rotated the letter p 180 counterclockwise, what letter would you have?
2. If you rotated the letter p 180 clockwise, what letter would you have?
3. A 90 clockwise rotation is the same as what counterclockwise rotation?
4. A 270 clockwise rotation is the same as what counterclockwise rotation?
5. A 200 counterclockwise rotation is the same as what clockwise rotation?
6. A 120 counterclockwise rotation is the same as what clockwise rotation?
7. A 340 counterclockwise rotation is the same as what clockwise rotation?
8. Rotating a figure 360 is the same as what other rotation?
9. Does it matter if you rotate a figure 180 clockwise or counterclockwise? Why or why not?
10. When drawing a rotated figure and using your protractor, would it be easier to rotate the figure 300
counterclockwise or 60 clockwise? Explain your reasoning.
(HONORS) From your notes, rotate each figure around point P the given angle measure.
11. 50
12. 120
13. 200
14. 330
15. 75
16. 170
For questions 17–25, rotate each figure counter-clockwise in the coordinate plane the given angle
measure. The center of rotation is the origin.
17. 180
18. 90
19. 180
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20. 270
21. 90
22. 270
23. 180
24. 270
25. 90
Find the measure of x in the rotations below.
26.
27.
28.
pre-image
5x + 2
17
7x - 3
100
210
pre-image
3x + 15
80
pre-image
2x - 9
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x + 12
Find the angle of rotation for the graphs below. The center of rotation is the origin. For each question,
the answer will be 90 , 270 , or 180 counter-clockwise.
29.
30.
31.
pre-image
pre-image
pre-image
32.
33.
pre-image
pre-image
The vertices of GHI are G(-2, 2), H(8, 2), and I(6, 8). Use this information to answer questions 34–37.
34. Plot GHI on the coordinate plane.
35. Reflect GHI over the x-axis. Find the coordinates of G ' H ' I ' .
36. Reflect G ' H ' I ' over the y-axis. Find the coordinates of
G '' H '' I '' .
37. What one transformation would be the same as this double
reflection?
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9.1g (apply)—Translation, Reflection, and Rotation Practice
TRANSLATION Practice:
Use the translation  x, y    x  2, y  4  for questions 1–7.
What are the coordinates of the image of
A  4, 1 ?
What are the coordinates of the image of
C  7, 2  ?
2.
What are the coordinates of the image of B  3,5 ?
4.
What are the coordinates of the pre-image of D ' ?
5.
What are the coordinates of the image of A ' ?
6.
What are the coordinates of the image of A '' ?
7.
Plot A, A ', A '' and A ''' from the questions above. What do you notice?
1.
3.
The vertices of ABC are A(-3, -5), B(-7, -4) and C(-6, 4). Find the vertices of ABC , given the
translation rules below.
8.
 x, y    x  4, y  2
9.
 x, y    x  8, y  5
10.
 x, y    x, y  2
11.
 x, y    x  4, y  5
12.
 x, y    x  2, y 
13.
 x, y    x  4, y  7 
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Reflection Practice
1. If (7, 2) is reflected over the y-axis, what are the coordinates of the image?
2. If (7, 2) is reflected over the x-axis, what are the coordinates of the image?
3. If (7, 2) is reflected over y  x , what are the coordinates of the image?
4. If (7, 2) is reflected over y   x , what are the coordinates of the image?
5. Plot the four images. What shape do they make? Be specific.
6. Which letter is a reflection over a vertical line of the letter b ?
7. Which letter is a reflection over a horizontal line of the letter b ?
Reflect each shape over the given line.
8. y-axis
9. x-axis
10. x  1
11. y  4
12. x-axis
13. y-axis
14. y  x
15. y   x
16. x  2
17. y  x
18. y   x
19. y  4
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Rotations Practice
For the point, line, or polygon given rotate it counter-clockwise for the given angle.
1. 180
2. 90
3. 180
4. 270
5. 90
6. 270
7. 180
8. 270
9. 90
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9.1h (apply)—Compositions of Transformations
1.
Explain why the composition of two or more isometries must also be an isometry. (Isometries are distance
preserving transformations.)
2.
What one transformation is the same as a reflection over two parallel lines?
3.
What one transformation is the same as a reflection over two intersecting lines?
Use the graph of the square to the right to answer questions 4–6.
4.
Perform a glide reflection (a combination of a reflection in a line and a
translation along the same line) over the x-axis and to the right 6 units.
Write the new coordinates.
5.
What is the rule for this glide reflection?
6.
What glide reflection would move the image back to the pre-image?
Use the graph of the square to the right to answer questions 7–9.
7.
Perform a glide reflection to the right 6 units, then over the x-axis. Write
the new coordinates.
8.
What is the rule for this glide reflection?
9.
Is the rule in #8 different than the rule in #5? Explain why or why not.
Use the graph of the triangle to the right to answer questions 10–12.
10. Perform a glide reflection over the y-axis and down 5 units. Write the
new coordinates.
11. What is the rule for this glide reflection?
12. What glide reflection would move the image back to the pre-image?
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Use the graph of the triangle to the right to answer questions 13–15.
13. Reflect the pre-image over the line y  1 followed by the line y  7 .
Draw the new triangle.
14. What one transformation is this double reflection the same as?
15. Write the rule.
Use the graph of the triangle to the right to answer questions 16–19.
16. Reflect the pre-image over the line y  7 followed by the line y  1 .
Draw the new triangle.
17. What one transformation is this double reflection the same as?
18. Write the rule.
19. How do the final triangles in #13 and #16 differ?
Use the trapezoid in the graph to the right to answer questions 20–22.
20. Reflect the pre-image over the x-axis then the y-axis. Draw the new
trapezoid.
21. Reflect the pre-image trapezoid over the y-axis then the x-axis. Draw
the new trapezoid.
22. Are the final trapezoids from #20 and #21 different? Explain why or
why not.
Extension: Answer the questions below. Be as specific as you can.
23. Two parallel lines are 7 units apart. If you reflect a figure over both how far apart will the pre-image and final image
be?
24. After a double reflection over parallel lines, a pre-image and its image are 28 units apart. How far apart are the
parallel lines?
25. Two lines intersect at a 165 angle. If a figure is reflected over both lines, how far apart will the pre-image and
image be?
26. What is the center of rotation for #25?
27. Two lines intersect at an 83 angle. If a figure is reflected over both lines, how far apart will the pre-image and
image be?
28. After a double reflection over parallel lines, a pre-image and its image are 62 units apart. How far apart are the
parallel lines?
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