Translations, Reflections, and Rotations

Vocabulary Transformation- changes the position or orientation of a figure. Image- the resulting figure after a transformation. Preimage- the original figure. Translation- figure slides along a straight line without turning. Reflection- figure flips across a line of reflection, creating a mirror image. Line of reflection- the line in which a figure flips across to create a mirror image. Rotation- figure turns around a fixed point.

In mathematics, a transformation changes the position or orientation of a figure. The resulting figure is the image of the original figure, called the preimage. Images resulting from the transformations described in the next slides are congruent to the original figures.

Translation The figure slides along a straight line without turning. Types of Transformations

Reflection The figure flips across a line of reflection, creating a mirror image. Types of Transformations

Rotation The figure turns around a fixed point. Types of Transformations

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.It is a reflection. The figure slides along a straight line.

Check It Out: Example 1 Identify the type of transformation.

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.

Additional Example 2 Continued Write the coordinate of the vertices of the image. The coordinates of the vertices of quadrilateral A'B'C'D' are A'(–3, 1), B'(0, 2), C'(0, –1), and D'(–3, –3). Quadrilateral ABCD(x – 4, y – 2)A’B’C’D’ A(1, 3)(1 – 4, 3 – 2)A’(–3, 1) B(4, 4)(4 – 4, 4 – 2)B’(0, 2) C(4, 1)(4 – 4, 1 – 2)C’(0, –1) D(1, –1)(1 – 4, –1 – 2)D’(–3, –3)

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

Check It Out: Example 2 Graph the translation of quadrilateral ABCD 5 units left and 3 units down.

Graph the reflection of the figure across the indicated axis. Write the coordinates of the vertices of the image. Additional Example 3: Graphing Reflections on a Coordinate Plane

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. The coordinates of the vertices of triangle A’D’C’ are A’( – 3, – 1), D’(0, 0), and 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. The coordinates of the vertices of triangle A’D’C’ are A’(3, 1), D’(0, 0), and C’( – 2, 2).

Check It Out: Example 3 Graph the reflection of quadrilateral ABCD across the x-axis.

Triangle ABC has vertices A(1, 0), B(3, 3), C(5, 0). Rotate ∆ABC 180° about the origin. Write the coordinates of the vertices of the image. Additional Example 4: Graphing Rotations on a Coordinate Plane 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’ The coordinates of the vertices of triangle A’B’C’ are A’(0, 0), B’(–2, –3), and C’(–4, 0).

Rotate the graph of quadrilateral ABCD 90° clockwise about the origin. Check It Out: Example 4