Understanding Translation Vectors

Oct 13, 2025 by wbrum.org 132 views

Introduction

In mathematics, the concept of translation is a fundamental aspect of geometry, where a figure is moved from one location to another without any change in its size or shape. In this article, we will explore the translation of a triangle with given coordinates and understand how its image is formed after the translation process.

Given Coordinates

The original triangle has coordinates ${$ F(-2, 3) $}$, ${$ G(-4, 1) $}$, and ${$ H(-2, -2) $}$. These coordinates represent the vertices of the triangle, and we can visualize the triangle in a coordinate plane.

Translation Process

The triangle is translated, and its image has coordinates ${$ F'(0, 0) $}$, ${$ G'(-2, -2) $}$, and ${$ H'(0, -5) $}$. The translation process involves moving each vertex of the original triangle to a new location, resulting in a new triangle with the given coordinates.

Understanding Translation Vectors

To understand the translation process, we need to identify the translation vectors that move each vertex from its original location to its new location. The translation vector is a vector that represents the change in the x and y coordinates of a point.

Translation Vector for F

The translation vector for point F is ${$ (0 - (-2), 0 - 3) = (2, -3) $}$. This means that point F is moved 2 units to the right and 3 units down to reach its new location.

Translation Vector for G

The translation vector for point G is ${$ (-2 - (-4), -2 - 1) = (2, -3) $}$. This means that point G is also moved 2 units to the right and 3 units down to reach its new location.

Translation Vector for H

The translation vector for point H is ${$ (0 - (-2), -5 - (-2)) = (2, -3) $}$. This means that point H is also moved 2 units to the right and 3 units down to reach its new location.

Common Translation Vector

We can see that all three translation vectors are the same, ${$ (2, -3) $}$. This means that the translation process involves moving each vertex of the original triangle 2 units to the right and 3 units down.

Image of the Triangle

The image of the triangle has coordinates ${$ F'(0, 0) $}$, ${$ G'(-2, -2) $}$, and ${$ H'(0, -5) $}$. We can visualize the image of the triangle by plotting these coordinates on a coordinate plane.

Conclusion

In conclusion, the translation of a triangle involves moving each vertex of the original triangle to a new location, resulting in a new triangle with the given coordinates. The translation process can be understood by identifying the translation vectors that move each vertex from its original location to its new location. By analyzing the translation vectors, we can see that the translation process involves moving each vertex 2 units to the right and 3 units down.

Real-World Applications

The concept of translation has many real-world applications in fields such as engineering, architecture, and computer graphics. For example, in computer-aided design (CAD), translation is used to move objects in a 3D space to create complex designs.

Mathematical Representation

The translation of a triangle can be represented mathematically using the following formula:

${$ (x', y') = (x + a, y + b) $}$

where ${$ (x, y) $}$ is the original coordinate, ${$ (x', y') $}$ is the new coordinate, and ${$ (a, b) $}$ is the translation vector.

Example Problems

A triangle has coordinates ${$ A(1, 2) $}$, ${$ B(3, 4) $}$, and ${$ C(5, 6) $}$. The triangle is translated, and its image has coordinates ${$ A'(2, 3) $}$, ${$ B'(4, 5) $}$, and ${$ C'(6, 7) $}$. Find the translation vector.
A triangle has coordinates ${$ D(-1, -2) $}$, ${$ E(-3, -4) $}$, and ${$ F(-5, -6) $}$. The triangle is translated, and its image has coordinates ${$ D'(0, -1) $}$, ${$ E'(-2, -3) $}$, and ${$ F'(-4, -5) $}$. Find the translation vector.

Solutions

The translation vector is ${$ (1, 1) $}$.
The translation vector is ${$ (1, 1) $}$.