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Geometric translation is a transformation that shifts a figure on a plane without altering its size, shape, or orientation. This text delves into the execution of translations using coordinate systems, formulating mathematical rules for precise movements, and implementing these on various shapes. It also provides real-world examples to illustrate the concept, highlighting the importance of maintaining the integrity of a figure's geometry during translation.

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## Definition of Translation

### Type of transformation

Translation is a transformation that shifts a figure on a plane without changing its size, shape, or orientation

### Pre-image and Image

The pre-image is the original figure before translation, and the image is the figure after translation

### Consistency of Translation

Translation involves moving every point of a figure consistently to maintain its intrinsic properties

## Execution of Translation

### Coordinate System

Translations are executed by altering the coordinates of a figure's points in a coordinate system

### Mathematical Rules

Specific mathematical rules govern the process of translation to ensure uniformity across the entire figure

### Formula for Translation

The general formula for translation is g(x, y) = (x + h, y + k), where h and k represent the horizontal and vertical shifts, respectively

## Types of Translation

### Horizontal Translation

Horizontal translation involves adding or subtracting a fixed value to the x-coordinates of a figure's points

### Vertical Translation

Vertical translation involves adding or subtracting a fixed value to the y-coordinates of a figure's points

### Combined Effect of Horizontal and Vertical Translation

When a figure undergoes both horizontal and vertical translations, the combined effect is represented by the function g(x, y) = (x + h, y + k)

## Application of Translation

### Adjusting Coordinates

To apply translation to a geometric shape, one must systematically adjust the coordinates of each vertex according to the translation vector's components

### Translation of Individual Points

The translation of individual points follows the same principles as those for entire shapes

### Real-World Examples

Real-world examples, such as translating a rectangle or triangle, effectively demonstrate the concept of translation

## Determining Translation

### Comparison of Coordinates

To determine the translation between a pre-image and its image, one must compare the coordinates of corresponding points

### Translation Vector

The difference in the x-coordinates gives the horizontal component of the translation, while the difference in the y-coordinates provides the vertical component

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