Transformations: Rotations & Dilations

A translation is a horizontal or vertical shift of a point or figure. The image produced by translation is congruent to the original.

A reflection is a mirror image of a point or figure taken across a reflection line. The reflections are usually across the x- or y-axis of a coordinate plane.

Yes; reflecting a figure does not change its angles or side lengths, so the image is congruent to the preimage.

Points are defined using an ordered pair (x, y) so that the first number is the distance from the origin on the x-axis, and the second is the distance from the origin on the y-axis.

Yes, I can perform multiple transformations to get an image. The order that I perform the transformations in matters, so I need to follow the order of transformations. This is similar to the order of operations. The order is rotations, reflections, dilations, and then translations.

A scale factor is the value I multiply the coordinates of the original point by to get its image point in a dilation.

Enlargements and reductions are both dilations, but an enlargement produces an image that is larger (farther from the origin) than the original, and a reduction produces an image that is smaller (closer to the origin). Enlargements have scale factors larger than 1, and reductions have scale factors between 0 and 1.

First, I need to know whether to rotate clockwise or counterclockwise. The mapping rule for a clockwise 90° rotation is (x,y)→(y,-x). If I want to rotate counterclockwise, I need to find the equivalent angle: 270° clockwise is the same as 90° counterclockwise. The mapping rule for a clockwise 270° rotation is (x,y)→(-y,x).

First, find the corresponding vertex points. Then determine if the image is in the same orientation or a different one. If the orientation is different, look for a rotation or reflection. If the orientation is the same, check if both figures are the same size. If they are not the same size, then a dilation was used. Divide the image coordinates by the corresponding original coordinates. The answer gives the scale factor for the dilation. If there was no dilation, then the difference between the image coordinates and the original coordinates gives the translation.

Two 2D figures are similar if all the corresponding angles are equal, and all side lengths share the same scale factor between them. Translations, reflections, dilations, and rotations all create similar shapes.