Translations, Reflections, and Rotations Course 2 8-10 Translations, Reflections, and Rotations In mathematics, a transformation changes the position or orientation of a figure. The resulting figure is the image of the original. Images resulting from the transformations described in the next slides are congruent to the original figures.

Types of Transformations Course 2 8-10 Translations, Reflections, and Rotations Types of Transformations Translation The figure slides along a straight line without turning.

Types of Transformations Course 2 8-10 Translations, Reflections, and Rotations Types of Transformations Reflection The figure flips across a line of reflection, creating a mirror image.

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.

Types of Transformations Course 2 8-10 Translations, Reflections, and Rotations Types of Transformations Rotation The figure turns around a fixed point.

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

A proportional Shrinking or enlargement of a figure Dilation: A proportional Shrinking or enlargement of a figure Under 1 will get smaller over 1 will get bigger

Additional Example 1: Identifying Types of Transformations Course 2 8-10 Translations, Reflections, and Rotations Additional Example 1: Identifying Types of Transformations Identify each type of transformation. A. B. The figure flips across the y-axis. The figure slides along a straight line. It is a reflection. It is a translation.

Insert Lesson Title Here Course 2 8-10 Translations, Reflections, and Rotations Insert Lesson Title Here The point that a figure rotates around may be on the figure or away from the figure. Helpful Hint

Insert Lesson Title Here Course 2 8-10 Translations, Reflections, and Rotations Insert Lesson Title Here Check It Out: Example 1 Identify each type of transformation. A. B. x y x y 4 4 2 2 –4 –2 2 4 –4 –2 2 4 –2 –2 –4 –4 The figure slides along a straight line. The figure turns around a fixed point. It is a translation. It is a rotation.

Additional Example 2: Graphing Transformations on a Coordinate Plane Course 2 8-10 Translations, Reflections, and Rotations 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.

Insert Lesson Title Here Course 2 8-10 Translations, Reflections, and Rotations Insert Lesson Title Here Reading Math A’ is read “A prime” and is used to represent the point on the image that corresponds to point A of the original figure

Insert Lesson Title Here Course 2 8-10 Translations, Reflections, and Rotations Insert Lesson Title Here Check It Out: Example 2 Translate quadrilateral ABCD 5 units left and 3 units down. x y B A 4 D’ C’ B’ A’ Each vertex is moved five units left and three units down. 2 C –4 –2 D 2 4 –2 –4

Additional Example 3: Graphing Reflections on a Coordinate Plane Course 2 8-10 Translations, Reflections, and Rotations Additional Example 3: Graphing Reflections on a Coordinate Plane 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 Continued Course 2 8-10 Translations, Reflections, and Rotations Additional Example 3 Continued A. x-axis. 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 ADC are A’(–3, –1), D’(0, 0), C’(2, –2).

Additional Example 3 Continued Course 2 8-10 Translations, Reflections, and Rotations Additional Example 3 Continued B. y-axis. 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 ADC are A’(3, 1), D’(0, 0), C’(–2, 2).

Insert Lesson Title Here Course 2 8-10 Translations, Reflections, and Rotations Insert Lesson Title Here Check It Out: Example 3A Graph the reflection of the triangle ABC across the x-axis. Write the coordinates of the vertices of the image. x y The x-coordinates of the corresponding vertices are the same, and the y-coordinates of the corresponding vertices are opposites. B 3 C A A’ B’ C’ 3 The coordinates of the vertices of triangle ABC are A’(1, 0), B’(3, –3), C’(5, 0). –3

Insert Lesson Title Here Course 2 8-10 Translations, Reflections, and Rotations Insert Lesson Title Here Check It Out: Example 3B Graph the reflection of the triangle ABC across the y-axis. Write the coordinates of the vertices of the image. x y B C 3 –3 The y-coordinates of the corresponding vertices are the same, and the x-coordinates of the corresponding vertices are opposites. C’ B’ A The coordinates of the vertices of triangle ABC are A’(0, 0), B’(–2, 3), C’(–2, –3).

Additional Example 4: Graphing Rotations on a Coordinate Plane Course 2 8-10 Translations, Reflections, and Rotations Additional Example 4: Graphing Rotations on a Coordinate Plane Triangle ABC has vertices A(1, 0), B(3, 3), C(5, 0). Rotate ∆ABC 180° about the vertex A. 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’

Translations, Reflections, and Rotations Course 2 8-10 Translations, Reflections, and Rotations Check It Out: Example 4 Triangle ABC has vertices A(0, –2), B(0, 3), C(0, –3). Rotate ∆ABC 180° about the vertex A. x y B The corresponding sides, AB and AB’ make a 180° angle. B’ C’ 3 A 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. 3 –3 C

Insert Lesson Title Here Course 2 8-10 Translations, Reflections, and Rotations Insert Lesson Title Here Lesson Quiz: Part I 1. Identify the transformation. reflection 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? (1, –4), (5, –4), (9, 4)

Insert Lesson Title Here Course 2 8-10 Translations, Reflections, and Rotations Insert Lesson Title Here 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. x y 2 –2 –4 4 C C’ B’ A’ B A C’’ A’’ B’’