In geometry, a transformation is a way of changing a shape's position, orientation, or size. Think of it as a set of instructions for moving a figure on a plane. The original figure is called the pre-image, and the new figure, after the move, is called the image.

There are four basic types of transformations we'll look at: translations, rotations, reflections, and dilations. The first three are known as rigid transformations or isometries because they don't change the shape's size or angles. They just move it around.

A translation is the simplest transformation. It just slides an object from one place to another. Every single point on the object moves the same distance in the same direction.

When you translate a shape, its size, angles, and orientation all stay exactly the same. The image is congruent to the pre-image. If you could pick it up, it would fit perfectly on top of the original.

A rotation turns a figure about a fixed point, called the center of rotation. To describe a rotation, you need two things: the center of rotation and the angle of rotation, which tells you how far to turn the figure. The angle is usually measured in degrees.

Like translations, rotations are rigid transformations. The rotated image is the same size and shape as the pre-image. However, its orientation—the direction it's facing—usually changes.

A reflection flips a figure across a line, called the line of reflection or the mirror line. Every point in the image is the same distance from the mirror line as the corresponding point in the pre-image, but on the opposite side.

Reflections preserve size and shape, so they are also isometries. But unlike translations and rotations, a reflection reverses the orientation of the figure. A right-hand glove, when reflected, becomes a left-hand glove.

Our final transformation is different. A dilation changes the size of a figure but not its shape. The figure can get bigger (an enlargement) or smaller (a reduction). A dilation is not a rigid transformation because it does not preserve distance.

A dilation has a center point and a scale factor. The scale factor, often written as $k$, tells you how much larger or smaller the object will become.

• If $k > 1$, the image is an enlargement.
• If $0 < k < 1$, the image is a reduction.
• If $k = 1$, the image is the same size as the pre-image.

Even though the size changes, the angles of the figure remain the same. The image is similar, but not congruent, to the pre-image.

Now that you've seen the four basic transformations, let's test your understanding.