Translation moves every point of a figure a fixed distance in the same direction, preserving the figure's shape and size. This rigid motion does not alter the orientation of the figure.

A reflection produces a mirror image of a figure across a specified line. The process reverses the orientation while preserving size and shape.

Rotation pivots a figure around a fixed point without altering its shape or size. All points of the figure maintain a constant distance from the center of rotation.

Dilation preserves angle measures, ensuring the image is similar to the original figure. While the size of the figure changes, its internal angles remain constant.

Translation involves moving every point of a figure the same distance in a given direction. It is a rigid motion that preserves both shape and size.

A 90° clockwise rotation sends (x, y) to (y, -x). This formula preserves the distance from the origin, maintaining the figure's congruence.

Reflection over the line y = x swaps the coordinates of a point. This transformation ensures the distance relative to the line remains maintained.

A glide reflection is achieved by reflecting a figure over a line and then translating it along that line. The specific order of these transformations is essential to create the glide reflection effect.

In a dilation, the area scales by the square of the scale factor. Therefore, a scale factor of 2 results in an area that is 2², or 4 times, the original.

Rotation maintains the distance of every point from the center of rotation, preserving the rigidity of the figure. Although the orientation of the figure changes, the distances from the center remain constant.

The line of reflection is the fundamental element needed for reflecting a figure. It serves as the mirror over which every point of the figure is reflected.

Rigid transformations preserve distances and angles, resulting in figures that are congruent to the original. They do not change the size or shape of the figure.

The function f(x, y) = (x + 3, y - 2) shifts every point by adding 3 to the x-coordinate and subtracting 2 from the y-coordinate, which is the definition of a translation. This is a straightforward example of a rigid motion.

Dilation changes the size of a figure by scaling distances relative to a fixed center point. In contrast, translations, rotations, and reflections are rigid motions that preserve distances.

Dilation multiplies each coordinate of the point by the scale factor. Multiplying (2, 3) by 0.5 yields (1, 1.5), which is the correct image of point P under the transformation.

A 180° rotation about any point, including the centroid, produces a congruent image with reversed orientation. While the location of the vertices changes, the size and shape of the triangle remain the same.

Reflecting (x, y) over the line y = 0 yields (x, -y). A subsequent 90° counterclockwise rotation transforms (x, -y) to (y, x) using the standard rotation rule. The composite transformation results in (y, x).

A dilation with a scale factor of 3 multiplies the distance from the center (2, -3) to every point on the figure by 3. The shape is enlarged while maintaining its overall proportions.

The area of a shape under dilation scales by the square of the scale factor. Since the area quadruples, the scale factor squared is 4, meaning the scale factor is 2.

Reflecting (3, -2) across the line x = 1 gives (2*1 - 3, -2) which is (-1, -2). Translating (-1, -2) upward by 5 units results in (-1, 3), the correct image of the point.