Reflections, Rotations, Translations LT 12.1, 12.2, 12.3

LT 12.1 Identify and Graph Reflections Recall that a reflection is a transformation that moves a figure (the preimage) by flipping it across a line. The reflected figure is called the image. A reflection is an isometry, so the image is always congruent to the preimage. Example 1: Identifying Reflections Tell whether each transformation appears to be a reflection. Explain. A. B.

Identify numbers that are… Reflections of itself across a horizontal line. Reflections of itself across a vertical line.

Reflections: flips over a given line, creates a mirror image. Over the x axis: (x, y) → (x, -y) Over the y axis: (x, y) → (-x, y) Over the line y = x: (x, y) → (y, x) Reflect over the x-axis Reflect over the y-axis Reflect over the line y=x (2, 3) (-3, 2) (6, -4) (-4, -4) (0, 9) (-7, 0)

Example 2 A city planner needs to connect Nealon City and Yencer City to Eiler Road. In order to save money he wants to build the roads as short as possible. In which city should the roads connect? Hint: Use a reflection. What is the shortest distance between two points? Nealon City Yencer City Eiler Road Angela City Clingman Town Ohringer City Quigle Town

LT 12.2 Identify and Graph Translations A translation is a transformation where all the points of a figure are moved the same distance in the same direction. A translation is an isometry, so the image of a translated figure is congruent to the preimage. Example 2: Identifying Translations Tell whether each transformation appears to be a translation. Explain. A. B.

Recall that a vector in the coordinate plane can be written as <a, b>, where a is the horizontal change and b is the vertical change from the initial point to the terminal point.

Example 3: Drawing Translations in the Coordinate Plane Translate the triangle with vertices D(–3, –1), E(5, –3), and F(–2, –2) along the vector <3, –1>.

Example 4 A rook on a chessboard has coordinates (3, 4). The rook is moved up two spaces. Then it is moved three spaces to the left. He did this 3 more times. What is the rook’s final position? What single vector moves the rook from its starting position to its final position?

Example 5 The vertices of a triangle are A(3, -3), B(3, 3), C(-1, 1). Create a vector that places the image of the triangle entirely in quadrant III.

LT 12.3 Identify and Graph Rotations Example 6: Identifying Rotations Tell whether each transformation appears to be a rotation. Explain. B. A.

Unless otherwise stated, all rotations in this book are counterclockwise. Helpful Hint

If the angle of a rotation in the coordinate plane is not a multiple of 90°, you can use sine and cosine ratios to find the coordinates of the image.

Example 7: Drawing Rotations in the Coordinate Plane Rotate ΔJKL with vertices J(2, 2), K(4, –5), and L(–1, 6) by 180° about the origin.

Example 8: Engineering Application A Ferris wheel has a 100 ft radius and takes 60 s to make a complete rotation. A chair starts at (100, 0). After 5 s, what are the coordinates of its location to the nearest tenth?