Warm Up 1. What is the translation rule? 2. What is the image after reflecting (2, 3) over the line y = x? 3. Identify the transformation of ABCD to OPQR.

Rotations & Symmetry

Rotations ROTATE x˚ about point Z. X = angle of rotation Z = Center of rotation This is an ISOMETRY

Rotations When you rotate an object (point), it will stay the SAME distance from the center of rotation.

Rotations On a graph: We will ALWAYS rotate about the origin We will only rotate 90˚ or 180˚

Use what we know… What do you know about 180˚? Straight line, straight angle So we will draw a line connecting the ORIGIN and our PREIMAGE point. (Extend past the origin.) Measure the distance from the ORIGIN to your PREIMAGE. Use this measurement to find the image point. The point equidistant from the origin is the IMAGE.

Rotate 180 ˚ You need to rotate each point individually 1. Connect the original point to the origin and extend 2. Measure the distance (preimage to origin) 3. Measure the extension (origin to equidistant) 4. Mark your image point What are the coordinates of S’U’N’? What do you notice about the coordinates?

90˚ rotations After you draw a line through the origin, use your PROTRACTOR to draw another line at 90˚ Notice the direction: clockwise or c ounterclockwise Measure the distance from the origin to the preimage Measure along the perpendicular line. The same distance from above will locate your image.

Rotate 90 ˚ You need to rotate each point individually 1. Connect the original point to the origin and extend 2. Draw in the perpendicular line 3. Measure the distance (preimage to origin) 4. Measure the perp. line (origin to equidistant) 5. Mark your image point What are the coordinates of A’B’C’D’? What do you notice about the coordinates?

Rotations 180˚ rotation: – negate each coordinate – So (2,4) becomes (-2, -4) 90˚ clockwise: – switch coordinates, negate new y-coordinate – So (2,4) becomes (4, -2) 90˚ counterclockwise: – switch coordinates, negate new x-coordinate – So (2,4) becomes (-4, 2)

Regular Polygons Regular: all sides are congruent and all angles are congruent 360˚ around center Degree measure of each angle around the center? 360 ÷ (# of sides) 72˚

Regular Polygons Rotate P: 144˚ to the right Rotate T: 216˚ to the right Rotate PE: 72˚ to the right A T N E P

Polygon Symmetry We can also say that the pentagon below can be rotated 72˚ in order to rotate the original pentagon onto itself. What other degrees of rotation can occur for the same result?

Polygon Symmetry We can also Reflect the pentagon over the red line and the pentagon will again appear to have been reflected onto itself. Are there any other lines that the pentagon can reflect over causing the same result?

Try a few more shapes… List the degrees of rotation that the shape can rotate onto itself. List the lines of symmetry that the shape can reflect onto itself.

Composition of Transformations A Composition of Transformations is the process of performing more than one transformation on a single preimage. (just like we did yesterday with the Glide Reflections) Some compositions you may run into: Glide Reflection Translation, then Rotation Rotation, then Reflection Or ALL THREE!

What do you notice? When you reflect over a set of two intersecting lines, the process is a composition. But if we look at our preimage and our final image… Its just a rotation! E E E E E

What happens here? Try to reflect over two PARALLEL lines. It’s a TRANSLATION! E E E EE

Homework Worksheet