Y. Davis Geometry Notes Chapter 9

Reflection A transformation that acts like a flip. A reflection in a line is a function that maps a point to its image such that: If the point is on the line, then the image and preimage are the same point. If the point is not on the line, the line is the perpendicular bisector of the segment joining the 2 points.

Line of Reflection The line that a reflection is flipped over. When reflecting in a line, find an image that is equidistant from the line, as the preimage. Reflection in x-axis— maps Reflection in y-axis— maps Reflection in y=x—maps

Translation A transformation that acts like a slide. Translations are mapped by vectors.

Rotation A transformation that acts like a turn about a fixed point. If the point is the center of rotation, then the image and preimage are the same point. If the point is not the center of rotation, then the image and preimage are the same distance from the center of rotation and the angle of rotation formed by the preimage, center of rotation, and image points is x.

Clockwise Rotations

Compostions of Transformations When 2 or more transformations are applied to a geometric figure.

Glide Reflection A translation followed by a reflection.

Theorem 9.1 Compostion of Isometries The composition of 2 or more isometries is an isometry.

Theorem 9.2 Reflections in Parallel Lines A composition of two reflections in parallel lines can be described by a translation vector that is perpendicular to the two lines, and twice the distance between the two lines A reflection in parallel lines is a translation.

Theorem 9.3 Reflections in intersecting lines The composition of two reflections in intersecting lines can be described by a rotation. about the point of intersection. through an angle that is twice the measure of the acute or right angel formed by the lines. A reflection in intersecting lines is a rotation.

Symmetry When a figure can be mapped onto itself by a reflection, rotation, translation or other rigid transformation.

Linear Symmetry (Reflection Symmetry) A line in a figure where it can be mapped onto itself by a reflection in the line.

Rotational Symmetry When a figure can be mapped onto itself by a rotation between 0 and 360 degrees. Order of symmetry--# of times a figure can be mapped onto itself. Magnitude of symmetry—the smallest angle measure that a figure can be rotated onto itself. Magnitude = 360 divided by order.

Plane Symmetry When a three-dimensional figure can be mapped onto itself by a reflection in a plane.

Axis Symmetry When a three-dimensional figure can be mapped onto itself by a rotation between 0 and 360 degrees in a line.

Dilations A transformation that enlarges or reduces the original figure proportionally.

Center of Dilation The fixed point that dilations are performed with respect to.

Scale factor of Dilation The ratio of a length on the image to a corresponding length on the preimage.

Enlargement A dilation with a scale factor greater than 1. (an image is larger than the preimage.)

Reduction A dilation with a scale factor between 0 & 1. (an image is smaller than the preimage.)