Section 2.2: Translations

Property

A transformation is a function that changes the position, size, or shape of a figure to create a new figure called the image. The four main types of transformations are: translation (slide), reflection (flip), rotation (turn), and dilation (resize).

Examples

Property

A translation is a rigid transformation that "slides" a figure across a plane to a new location. Every single point of the original figure (the pre-image) moves the exact same distance and in the exact same direction to create the new figure (the image). Because it is a rigid motion, the figure does not rotate, reflect, or change its size. Therefore, the pre-image and image are perfectly congruent and face the exact same way (they preserve orientation).

Examples

• Macro View: Sliding a physical ruler across your desk without rotating it.
• Micro Detail (Naming): When triangle ABC slides to a new position, the new triangle is named A'B'C' (read as "A prime, B prime, C prime"). Point A matches with A', B with B', and C with C'.
• Micro Detail (Direction): If you draw a straight line from A to A' and another from B to B', those lines will be perfectly parallel and the exact same length.

Explanation

While the property tells us the shape just "slides," here are the micro-details to watch out for:

1. Pre-image vs. Image: The original starting shape is called the "pre-image" (usually standard letters like A, B, C). The final landing spot is the "image" (indicated by the prime marks like A', B', C').
2. Congruence: Because it's a "rigid" motion, the pre-image and image are exactly identical. If the side length of AB was 5 units, the side length of A'B' is strictly 5 units. No stretching allowed!

Property

The axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise.

Points on the Axes
Points with a $y$-coordinate equal to 0 are on the $x$-axis, and have coordinates $(a, 0)$.
Points with an $x$-coordinate equal to 0 are on the $y$-axis, and have coordinates $(0, b)$.