Transformations in Geometry

A translation slides a figure along a straight line without changing its size or shape. It is defined by a vector that indicates both the direction and distance of the slide. Every point of the figure moves the same distance and in the same direction. A translation can be described by a rule that shows how each coordinate point is shifted, such as (x, y) → (x+3, y-2).

The scale factor in a dilation determines the amount of enlargement or reduction of the figure. A scale factor greater than 1 results in an enlargement, while a scale factor between 0 and 1 results in a reduction. The size of the image is directly proportional to the scale factor; a larger scale factor produces a larger image, and vice versa.

A rotation turns a figure around a fixed point, the center of rotation, by a specific angle. It preserves the size and shape of the figure, while changing its orientation. It is defined by the center of rotation and the angle of rotation. A reflection flips a figure over a line, the line of reflection. It also preserves the size and shape but creates a mirror image of the original figure. It is defined by the line of reflection.

<p>Isometries are transformations that preserve distances and angles. This means they create congruent figures, with the same size and shape as the original. Examples of isometries include translations, rotations, and reflections. Dilations are not isometries because they change the size of the figure, even though they preserve angles.</p> Signup and view all the answers

<p>In a dilation, a point on the figure and its corresponding point on the image are collinear with the center of dilation. The distance from the center of dilation to the image point is equal to the product of the scale factor and the distance from the center of dilation to the original point.</p> Signup and view all the answers

<p>The image of triangle ABC after the translation would be triangle A'B'C' with vertices A'(5,-1), B'(7,1), and C'(4,3). Each coordinate is shifted 3 units to the right and 2 units down.</p> Signup and view all the answers

<p>Transformations are used in a variety of real-world applications, especially in computer graphics and design. For instance, image editing software uses transformations like translations, rotations, and reflections to resize, rotate, or flip images. Similarly, computer animation relies on transformations to move and manipulate objects in virtual environments, creating dynamic and realistic effects.</p> Signup and view all the answers

<p>Dilations preserve angle measures. This is because dilations only change the size of a figure, not its shape. Angles are determined by the relative positions of lines, not by the length of the sides. Since the relative positions of lines remain unchanged in a dilation, the angles remain the same.</p> Signup and view all the answers

Functions that move or change a geometric figure in specific ways.

A slide of a figure along a straight line without changing size or shape.

A quantity that defines the direction and distance of a translation.

An enlargement or reduction of a figure by a scale factor around a fixed center.

Determines how much a figure is enlarged or reduced in a dilation.

Turns a figure around a fixed point at a specific angle.

Flips a figure over a line of reflection acting as a mirror.

Transformations that preserve distances and angles; include translations, rotations, and reflections.