Question
Define translation in geometry.
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GPT-4o mini
In geometry, a translation refers to a transformation that moves a figure or object from one location to another without altering its shape, size, or orientation. This movement is characterized by shifting every point of the figure by the same distance in the same direction.
Mathematically, a translation can be represented using vector notation. For example, if a point \( P(x, y) \) is translated by a vector \( \mathbf{v} = (a, b) \), the new position of the point \( P' \) after the translation would be given by:
\[
P'(x', y') = (x + a, y + b)
\]
Here, \( a \) is the horizontal movement and \( b \) is the vertical movement. The key aspects of translation include:
1. **Preservation of shape and size**: The translated figure is congruent to the original.
2. **Direction and distance**: All points are moved the same distance in the same direction.
3. **One-to-one correspondence**: Each point of the original figure corresponds uniquely to a point in the translated figure.
Translations are commonly used in various branches of mathematics and applications, including computer graphics and physical simulations.
Mathematically, a translation can be represented using vector notation. For example, if a point \( P(x, y) \) is translated by a vector \( \mathbf{v} = (a, b) \), the new position of the point \( P' \) after the translation would be given by:
\[
P'(x', y') = (x + a, y + b)
\]
Here, \( a \) is the horizontal movement and \( b \) is the vertical movement. The key aspects of translation include:
1. **Preservation of shape and size**: The translated figure is congruent to the original.
2. **Direction and distance**: All points are moved the same distance in the same direction.
3. **One-to-one correspondence**: Each point of the original figure corresponds uniquely to a point in the translated figure.
Translations are commonly used in various branches of mathematics and applications, including computer graphics and physical simulations.