Transformations: Translations, Rotations, and Reflections

**Transformations**

Understanding the concepts of simple geometric transformations **– translations, rotations, and reflections** will help you work through some of the math questions.

A **translation** moves a shape without any rotation or reflection. For example, the square on the left has been translated 2 units up (that is, in the positive *y*-direction) to get the square on the right.

**Rotating** an object means turning it around a point, which is called the **center of rotation**. For example, when the clock face on the left is rotated 90 counterclockwise, the result is the clock face on the right:

To **reflect** an object means to produce its mirror image with respect to a line, which is called the **line of reflection**. The figure below shows a triangle reflected across the line *l*. A mirror image is produced on the other side of the line.

If you reflect a figure twice across the same line, you get back the original figure.

## Symmetry

Look at the figure below to review the concept of symmetry:

The dashed line *m* divides the hexagon into two halves. If the left half is reflected across line *m*, the result is the right half- and vice versa. In other words, if you reflect the hexagon across line *m*, the result is the same hexagon. The hexagon is said to be symmetrical about the line *m*, and *m* is called a **line of symmetry** for the hexagon.

A geometric figure may have more than one line of symmetry (for example, a rectangle), or it may have no lines of symmetry.

There is another type of symmetry. Look at the rectangle below, with point *P* at its center.

If you rotate the rectangle 180 (clockwise or counterclockwise) around *P*, then the result is the same rectangle. The rectangle is said to be symmetric around the point *P*, and *P* is called a **point of symmetry** for the rectangle.

Note that symmetry about a line and symmetry about a point are different properties. A given figure may have either type of symmetry and not the other, or it may have both types of symmetry (for example, a circle) or neither type.