In mathematics, a transformation changes the position or orientation of a figure. The resulting figure is the image of the original. Images resulting from.

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

TranslationThe figure slides along a straight line without turning.

Course 2

8-10 Translations, Reflections, and Rotations

Types of Transformations

ReflectionThe figure flips across a line of reflection, creating a mirror image.

Course 2

8-10 Translations, Reflections, and Rotations

Types of Transformations

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.

RotationThe figure turns around a fixed point.

Course 2

8-10 Translations, Reflections, and Rotations

Types of Transformations

Angle of Rotation

The measure of degrees that a figure is rotated around a fixed point

Dilation: A proportional Shrinking or

enlargement of a figureUnder 1 will get smaller

over 1 will get bigger

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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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

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.

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

Insert Lesson Title Here

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

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

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

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).

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).

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’

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’

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’

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

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

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’’