Translation shifts a triangle in a fixed direction without altering its orientation or size. It simply slides the triangle from one location to another.

Rotation turns a triangle around a fixed point, changing its orientation but leaving its shape and size intact. This is a standard rigid motion.

Reflection flips a triangle over a designated line, creating a mirror image of the original figure. It reverses the orientation of the triangle while keeping its dimensions the same.

A dilation enlarges or reduces the triangle while preserving its shape and proportionality of its sides. The transformation changes the size but not the overall structure of the triangle.

Translation moves the triangle without altering its size, orientation, or shape. It simply shifts the position of the triangle on the plane.

A 180-degree rotation produces a triangle that is congruent to the original because every point is mapped to an opposite location. The overall shape and size remain unchanged despite the change in orientation.

Reflecting over the y-axis negates the x-coordinate and leaves the y-coordinate unchanged. This is a standard result of reflection across the y-axis.

A 90-degree clockwise rotation swaps the coordinates and negates the original x-coordinate, resulting in (y, -x). This rule is commonly applied in coordinate geometry for such rotations.

Rotation turns a triangle around a fixed point without producing a mirror image; it simply changes the triangle's orientation. This transformation is distinct from reflection, which creates a reversed image.

Dilation with a scale factor of 2 enlarges the triangle so that each side is twice as long as before. The shape remains similar but its size is increased uniformly.

Translation moves the triangle without affecting its dimensions, while dilation scales the triangle, thereby altering its size. This difference is crucial when distinguishing between rigid and non-rigid transformations.

Reflecting over the line y = x interchanges the x and y coordinates of each vertex. This simple swap is a key property of reflection over the line y = x.

Rotation is classified as a rigid transformation because it preserves both side lengths and angles in a triangle. This means the triangle remains congruent to its original form despite changes in orientation.

Dilation changes the size of a triangle, creating an image that is similar but not necessarily congruent to the original. In contrast, translations, rotations, and reflections are rigid transformations that preserve size.

Since translations, rotations, and reflections are all rigid transformations, they preserve both size and shape. Therefore, the triangles remain congruent after these transformations.

Reflecting over the line y = -x swaps the coordinates and negates both, resulting in the transformation (x, y) → (-y, -x). This rule is essential when dealing with reflections across lines not parallel to the axes.

First, the dilation scales (x, y) to (x/2, y/2). Then, a 90-degree counterclockwise rotation transforms (x/2, y/2) to (-y/2, x/2). This sequential approach demonstrates how combined transformations affect coordinates.

Reflecting (x, y) across the x-axis yields (x, -y). The subsequent translation adds 3 to the y-coordinate, resulting in (x, -y + 3). The order of operations is critical in composite transformations.

To determine the final position, one must first perform the rotation about the specified point on all vertices and then apply the translation vector. This sequential application of transformations ensures accurate mapping of the triangle's image.

Dilating triangle PQR with a scale factor of 3 multiplies each side length by 3, turning 3 to 9, 4 to 12, and 5 to 15. This maintains the triangle's shape but adjusts its size.