in mathematics, a transformation changes the position or orientation of a figure. the resulting...
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In mathematics, a transformationchanges the position or orientation of a figure. The resulting figure is the imageof the original. Images resulting fromthe transformations described in the next slides are congruent to the original figures.
Course 2
8-10 Translations, Reflections, and Rotations
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TranslationThe figure slides along a straight line without turning.
Course 2
8-10 Translations, Reflections, and Rotations
Types of Transformations
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ReflectionThe figure flips across a line of reflection, creating a mirror image.
Course 2
8-10 Translations, Reflections, and Rotations
Types of Transformations
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Reflection Line
is a line that lies in a position between two identical mirror images so that any point on one image is the same distance from the line as the same point on the other flipped image.
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RotationThe figure turns around a fixed point.
Course 2
8-10 Translations, Reflections, and Rotations
Types of Transformations
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Angle of Rotation
The measure of degrees that a figure is rotated around a fixed point
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Dilation: A proportional Shrinking or
enlargement of a figureUnder 1 will get smaller
over 1 will get bigger
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Identify each type of transformation.
Additional Example 1: Identifying Types of Transformations
The figure flips across the y-axis.
A. B.
It is a translation.Course 2
8-10 Translations, Reflections, and Rotations
It is a reflection.
The figure slides along a straight line.
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Insert Lesson Title Here
Course 2
8-10 Translations, Reflections, and Rotations
The point that a figure rotates around may be on the figure or away from the figure.
Helpful Hint
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Check It Out: Example 1
Identify each type of transformation.
A. B.
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Course 2
8-10 Translations, Reflections, and Rotations
x
y
2
2
–2
–4
4
4
–4
–2 0
x
y
2
2
–2
–4
4
4
–4
–2 0
It is a translation.
The figure slides along a straight line.
It is a rotation.
The figure turns around a fixed point.
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Additional Example 2: Graphing Transformations on a Coordinate Plane
Graph the translation of quadrilateral ABCD 4 units left and 2 units down.
Each vertex is moved 4 units left and 2 units down.
Course 2
8-10 Translations, Reflections, and Rotations
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A’ is read “A prime” and is used to represent the point on the image that corresponds to point A of the original figure
Reading Math
Course 2
8-10 Translations, Reflections, and Rotations
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Check It Out: Example 2
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Translate quadrilateral ABCD 5 units left and 3 units down.
Each vertex is moved five units left and three units down.
x
yA
B
C
2
2
–2
–4
4
4
–4
–2 D
D’C’
B’A’
Course 2
8-10 Translations, Reflections, and Rotations
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Graph the reflection of the figure across the indicated axis. Write the coordinates of the vertices of the image.
x-axis, then y-axis
Additional Example 3: Graphing Reflections on a Coordinate Plane
Course 2
8-10 Translations, Reflections, and Rotations
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A. x-axis.
Additional Example 3 Continued
The x-coordinates of the corresponding vertices are the same, and the y-coordinates of the corresponding vertices are opposites.
Course 2
8-10 Translations, Reflections, and Rotations
The coordinates of the vertices of triangle ADC are A’(–3, –1), D’(0, 0), C’(2, –2).
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B. y-axis.
Additional Example 3 Continued
The y-coordinates of the corresponding vertices are the same, and the x-coordinates of the corresponding vertices are opposites.
Course 2
8-10 Translations, Reflections, and Rotations
The coordinates of the vertices of triangle ADC are A’(3, 1), D’(0, 0), C’(–2, 2).
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Check It Out: Example 3A
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3
x
y
A
B
C
3
–3
Course 2
8-10 Translations, Reflections, and Rotations
Graph the reflection of the triangle ABC across the x-axis. Write the coordinates of the vertices of the image.
The x-coordinates of the corresponding vertices are the same, and the y-coordinates of the corresponding vertices are opposites.
The coordinates of the vertices of triangle ABC are A’(1, 0), B’(3, –3), C’(5, 0).
A’
B’
C’
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Check It Out: Example 3B
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A x
y
B
C
3
3
–3
Course 2
8-10 Translations, Reflections, and Rotations
Graph the reflection of the triangle ABC across the y-axis. Write the coordinates of the vertices of the image.
The y-coordinates of the corresponding vertices are the same, and the x-coordinates of the corresponding vertices are opposites.
The coordinates of the vertices of triangle ABC are A’(0, 0), B’(–2, 3), C’(–2, –3).C’
B’
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Triangle ABC has vertices A(1, 0), B(3, 3), C(5, 0). Rotate ∆ABC 180° about the vertex A.
Additional Example 4: Graphing Rotations on a Coordinate Plane
Course 2
8-10 Translations, Reflections, and Rotations
x
y
A
B
C
3
–3
The corresponding sides, AC and AC’ make a 180° angle.
Notice that vertex C is 4 units to the right of vertex A, and vertex C’ is 4 units to the left of vertex A.
C’
B’
A’
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Triangle ABC has vertices A(0, –2), B(0, 3), C(0, –3). Rotate ∆ABC 180° about the vertex A.
Check It Out: Example 4
Course 2
8-10 Translations, Reflections, and Rotations
The corresponding sides, AB and AB’ make a 180° angle.
Notice that vertex B is 2 units to the right and 3 units above vertex A, and vertex B’ is 2 units to the left and 3 units below vertex A.
x
y
B
C
3
3
–3B’
C’
A
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Lesson Quiz: Part I
1. Identify the transformation.
(1, –4), (5, –4), (9, 4)
reflection
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2. The figure formed by (–5, –6), (–1, –6), and (3, 2) is transformed 6 units right and 2 units up. What are the coordinates of the new figure?
Course 2
8-10 Translations, Reflections, and Rotations
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Lesson Quiz: Part II
3. Graph the triangle with vertices A(–1, 0), B(–3, 0), C(–1, 4). Rotate ∆ABC 90° counterclockwise around vertex B and reflect the resulting image across the y-axis.
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Course 2
8-10 Translations, Reflections, and Rotations
x
y
2
–2
2–2–4
–4
4
4
C
B AC’
B’
A’
C’’A’’
B’’