Finding the Image Coordinates

Oct 13, 2025 by miodeya.com 35 views

Introduction

In geometry, translation is a fundamental concept that involves moving a shape or a figure from one position to another without changing its size or orientation. When a triangle is translated, its image is formed by moving each of its vertices to a new location. In this article, we will explore how to find the image coordinates of a triangle after it has been translated.

Understanding Translation

Translation is a type of transformation that involves moving a shape or a figure from one position to another. It is a rigid motion, meaning that the shape or figure remains unchanged in size and orientation. The only thing that changes is its position. In the case of a triangle, translation involves moving each of its vertices to a new location.

The Original Triangle

The original triangle has coordinates F(-2,3), G(-4,1), and H(-2,-2). To find the image coordinates of the triangle after translation, we need to understand how the translation affects each of the vertices.

The Translation

The translation involves moving each of the vertices of the triangle to a new location. The new coordinates of the vertices are F'(0,0), G'(-2,-2), and H'(0,-5). To find the image coordinates of the triangle, we need to understand how the translation affects each of the vertices.

Finding the Image Coordinates

To find the image coordinates of the triangle, we need to apply the translation to each of the vertices. The translation can be represented by a vector, which is a mathematical object that has both magnitude and direction. In this case, the translation vector is (2,2), which means that each of the vertices is moved 2 units to the right and 2 units up.

Applying the Translation

To apply the translation to each of the vertices, we need to add the translation vector to each of the coordinates. For the vertex F, the new coordinates are F'(0,0) = (-2+2, 3+2) = (0,5). For the vertex G, the new coordinates are G'(-2,-2) = (-4+2, 1+2) = (-2,3). For the vertex H, the new coordinates are H'(0,-5) = (-2+2, -2+2) = (0,-5).

Conclusion

In conclusion, the image coordinates of the triangle after translation are F'(0,0), G'(-2,-2), and H'(0,-5). The translation involved moving each of the vertices of the triangle to a new location, resulting in a new triangle with the same size and orientation as the original triangle.

Example Problems

Here are a few example problems that involve finding the image coordinates of a triangle after translation:

A triangle has coordinates A(1,2), B(3,4), and C(5,6). The triangle is translated by a vector (2,3). What are the image coordinates of the triangle?
A triangle has coordinates D(-1,1), E(-3,3), and F(-5,5). The triangle is translated by a vector (4,5). What are the image coordinates of the triangle?
A triangle has coordinates G(2,1), H(4,3), and I(6,5). The triangle is translated by a vector (1,2). What are the image coordinates of the triangle?

Solutions

Here are the solutions to the example problems:

The image coordinates of the triangle are A'(3,5), B'(5,7), and C'(7,9).
The image coordinates of the triangle are D'(3,6), E'(1,8), and F'(-1,10).
The image coordinates of the triangle are G'(3,3), H'(5,5), and I'(7,7).

Tips and Tricks

Here are a few tips and tricks for finding the image coordinates of a triangle after translation:

Make sure to understand the translation vector and how it affects each of the vertices.
Use the translation vector to add to each of the coordinates of the vertices.
Check your work by plugging in the new coordinates into the original equation.

Real-World Applications

Translation is an important concept in geometry that has many real-world applications. Here are a few examples:

Architecture: Translation is used in architecture to move buildings and other structures from one location to another.
Engineering: Translation is used in engineering to move machines and other equipment from one location to another.
Computer Graphics: Translation is used in computer graphics to move objects and other graphics from one location to another.

Conclusion

In conclusion, translation is an important concept in geometry that involves moving a shape or a figure from one position to another without changing its size or orientation. The image coordinates of a triangle after translation can be found by applying the translation vector to each of the vertices. With practice and patience, you can become proficient in finding the image coordinates of a triangle after translation.