Pre Algebra Transformations

Transformations Intro to Rigid Transformations ● A transformation is a change in the position, size, or orientation of a figure. ○ The original figure is called the preimage. ○ The resulting (new) figure is called the image. ○ A transformation maps the preimage to the image. ● Rigid Transformations: the preimage and the image are congruent, which means they are the same size and shape. ● Arrow notation is used to describe a transformation, and prime notation is used to label the image. ● Reflection - A transformation across a line, called the line of reflection. ○ Each point and its image are the same distance from the line of reflection. ■ (Orientation changes!) ● Rotation - A transformation about a point P, called the center of rotation. ○ Each point and its image are the same distance from P. ● Translation - A transformation in which all points of a figure move the same distance in the same direction. Translations ● A translation is a transformation that slides a figure of fixed distance in a given direction. ● In a translation, all of the points move the same distance in the same direction. ● A translation is called a rigid transformation because the image is congruent to the preimage (which means same size and shape). ● In a translation of triangle ABC to triangle A’B’C, the distances from the pre-image points to the image points are equal, and the segments representing these distances are parallel. ○ The preimage and the image are congruent since line segments are the same length (same size) and corresponding angles have the same measure (same shape) in the preimage and image. ● Properties preserved under a translation from the pre-image to the image: 1. Distance 2. Angle measures 3. Parallelism 4. Orientation Reflections ● A reflection is a rigid transformation in which a figure is reflected or flipped across a line called the line of reflection. ● Each point of the preimage and image is equally distant from the line of reflection. ○ The preimage and the image are congruent because in a rigid transformation, the position of a figure changes but size and shape are preserved. ● After a reflection, the image is a mirror image of the pre-image; therefore, the orientation is changed. Rotations ● A rotation is a rigid transformation that turns a figure about a fixed point called the center of rotation a given number of degrees called the angle of rotation. ● The preimage and image are the same shape and size but the figures may be turned in different directions. ● A rotation maps a line to line, a ray to ray, a segment to segment, and an angle to angle. A rotation preserves lengths and segments, and a rotation preserves measures of a corresponding angle. Dilations on the Coordinate Plane ● A dilation is a transformation in which a figure is enlarged or reduced by a scale factor with respect to a fixed point (often the origin). ● The preimage and the image are similar but not congruent. Congruent vs. Similar Figures ● Congruent Figures: Two figures are congruent if one can be obtained from the other by a sequence of rotations, reflections and translations (rigid transformations). ● Similar figures: Two figures are similar to each other if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations. ○ They have the same shape but different sizes. Similar Triangles- Angle-Angle Criterion and Scale Factor/Side Lengths ● Two triangles are similar if they have congruent corresponding angles and proportional corresponding sides. ● If two triangles are congruent, their corresponding angles and corresponding sides will be congruent. Angle-Angle (AA) Similarity Postulate ● If two triangles have two pairs of corresponding angles, then they are similar triangles. Similar Triangles- Scale Factor and Side Lengths ● If we don't have info about corresponding angles (two pairs), we need to examine the ratios of the corresponding side lengths. This will help us prove the triangles are similar. ○ If the ratios of corresponding side lengths are equal, then the triangles are similar, meaning the lengths of the triangles were dilated by the same scale factor. ■ If two triangles are similar, comparing the ratios of corresponding sides will produce a constant scale factor.