Translation in Mathematics

Translation is a term used in mathematics to describe the movement of an object from one location to another without rotating or changing its size. In geometry, a translation refers to shifting a figure along a straight line in a plane. This movement can be in any direction - left, right, up, down, or even diagonally.

Translation Notation

When describing a translation, we use the notation T_(a,b), where (a,b) represents the amount of horizontal and vertical movement, respectively. If a is positive, the figure moves to the right; if a is negative, the figure moves to the left. Similarly, if b is positive, the figure moves upwards, and if b is negative, the figure moves downwards.

Vector Notation

Another way to describe a translation is using vector notation. A vector v can be used to represent the direction and distance of the translation. For example, if the vector v = (3, 4), it means the figure is translated 3 units to the right and 4 units up.

Properties of Translations

Some important properties of translations include:

• Translations preserve the size and shape of the original figure.
• Corresponding points on the original figure and its image are the same distance apart.
• Translations are a type of isometry, which means they preserve distances and angles.

Practice Problems

Let's try some practice problems to reinforce your understanding of translations:

1. Translate the point (2, 5) using the translation T_{(-3, 1)}.
2. If a figure is translated using the vector v = (-4, 2), in which direction and how far does the figure move?
3. What are the properties of a translation?

Answers

1. The translated point is (-1, 6). To find this, we subtract the horizontal movement from the x-coordinate and add the vertical movement to the y-coordinate.
2. The figure moves 4 units to the left and 2 units up.
3. The properties of a translation include preserving size and shape, maintaining distances between corresponding points, and being a type of isometry.