Triangle EFG Has Vertices \[$E(-3,4), F(-5,-1),\$\] And \[$G(1, 1).\$\] The Triangle Is Translated So That The Coordinates Of The Image Are \[$E^{\prime}(-1,0), F^{\prime}(-3,-5),\$\] And \[$G^{\prime}(3,-3).\$\]Which

Feb 27, 2025 by ADMIN 218 views

Triangle Translation: Understanding the Concept and Calculating the Image Coordinates

In geometry, translation is a fundamental concept that involves moving a figure from one location to another without changing its size or shape. This article will focus on the translation of a triangle, specifically the triangle EFG, and explore how to calculate the image coordinates of its vertices after translation.

Translation is a type of transformation that moves a figure from one location to another. It is a rigid motion, meaning that the size and shape of the figure remain unchanged. In the context of the triangle EFG, translation involves moving each vertex of the triangle to a new location while maintaining its original shape and size.

To calculate the image coordinates of the triangle EFG after translation, we need to understand the concept of translation vectors. A translation vector is a vector that represents the movement of a point from its original location to its new location. In this case, we are given the coordinates of the original vertices of the triangle EFG and the coordinates of their images after translation.

Translation Vectors

A translation vector can be represented as (h, k), where h is the horizontal component and k is the vertical component. In the case of the triangle EFG, we can calculate the translation vectors for each vertex as follows:

For vertex E(-3, 4), the translation vector is (-1 - (-3), 0 - 4) = (2, -4)
For vertex F(-5, -1), the translation vector is (-3 - (-5), -5 - (-1)) = (2, -4)
For vertex G(1, 1), the translation vector is (3 - 1, -3 - 1) = (2, -4)

Calculating the Image Coordinates

Now that we have the translation vectors, we can calculate the image coordinates of the vertices of the triangle EFG. To do this, we add the translation vector to the original coordinates of each vertex.

For vertex E(-3, 4), the image coordinates are (-3 + 2, 4 - 4) = (-1, 0)
For vertex F(-5, -1), the image coordinates are (-5 + 2, -1 - 4) = (-3, -5)
For vertex G(1, 1), the image coordinates are (1 + 2, 1 - 4) = (3, -3)

In conclusion, translation is a fundamental concept in geometry that involves moving a figure from one location to another without changing its size or shape. By understanding the concept of translation vectors and calculating the image coordinates of the vertices of a triangle, we can determine the new location of the triangle after translation. This article has provided a step-by-step guide on how to calculate the image coordinates of the triangle EFG after translation, using the given coordinates of the original vertices and the coordinates of their images.

A triangle has vertices at (2, 3), (4, 5), and (6, 7). If the triangle is translated 3 units to the left and 2 units up, what are the coordinates of the image vertices?
A triangle has vertices at (-2, 1), (-4, 3), and (-6, 5). If the triangle is translated 2 units to the right and 1 unit down, what are the coordinates of the image vertices?

For the first problem, the translation vector is (-3, 2). Adding this vector to the original coordinates of each vertex, we get:
- (2 - 3, 3 + 2) = (-1, 5)
- (4 - 3, 5 + 2) = (1, 7)
- (6 - 3, 7 + 2) = (3, 9)
For the second problem, the translation vector is (2, -1). Adding this vector to the original coordinates of each vertex, we get:
- (-2 + 2, 1 - 1) = (0, 0)
- (-4 + 2, 3 - 1) = (-2, 2)
- (-6 + 2, 5 - 1) = (-4, 4)

Translation has numerous applications in various fields, including:

Computer Graphics: Translation is used to move objects in 2D and 3D space, creating the illusion of movement and animation.
Architecture: Translation is used to design and build structures, such as buildings and bridges, by moving and scaling objects.
Engineering: Translation is used to design and build machines and mechanisms, such as gears and linkages, by moving and scaling objects.

In our previous article, we explored the concept of translation in geometry and calculated the image coordinates of the triangle EFG after translation. In this article, we will answer some frequently asked questions about triangle translation to help you better understand this concept.

Q: What is translation in geometry?

A: Translation is a type of transformation that moves a figure from one location to another without changing its size or shape.

Q: How do I calculate the image coordinates of a triangle after translation?

A: To calculate the image coordinates of a triangle after translation, you need to understand the concept of translation vectors. A translation vector is a vector that represents the movement of a point from its original location to its new location. You can calculate the translation vector by subtracting the original coordinates of the vertex from the new coordinates of the vertex.

Q: What is a translation vector?

A: A translation vector is a vector that represents the movement of a point from its original location to its new location. It can be represented as (h, k), where h is the horizontal component and k is the vertical component.

Q: How do I add a translation vector to the original coordinates of a vertex?

A: To add a translation vector to the original coordinates of a vertex, you simply add the horizontal component of the translation vector to the x-coordinate of the vertex and add the vertical component of the translation vector to the y-coordinate of the vertex.

Q: What are some real-world applications of translation?

A: Translation has numerous applications in various fields, including computer graphics, architecture, and engineering. In computer graphics, translation is used to move objects in 2D and 3D space, creating the illusion of movement and animation. In architecture, translation is used to design and build structures, such as buildings and bridges, by moving and scaling objects. In engineering, translation is used to design and build machines and mechanisms, such as gears and linkages, by moving and scaling objects.

Q: Can I use translation to scale a figure?

A: No, translation is used to move a figure from one location to another without changing its size or shape. If you want to scale a figure, you need to use a different type of transformation, such as dilation or scaling.

Q: Can I use translation to rotate a figure?

A: No, translation is used to move a figure from one location to another without changing its size or shape. If you want to rotate a figure, you need to use a different type of transformation, such as rotation.

Q: Can I use translation to reflect a figure?

A: No, translation is used to move a figure from one location to another without changing its size or shape. If you want to reflect a figure, you need to use a different type of transformation, such as reflection.

In conclusion, translation is a fundamental concept in geometry that involves moving a figure from one location to another without changing its size or shape. By understanding the concept of translation vectors and calculating the image coordinates of the vertices of a triangle, we can determine the new location of the triangle after translation. This article has provided a Q&A guide to help you better understand the concept of triangle translation.

A triangle has vertices at (2, 3), (4, 5), and (6, 7). If the triangle is translated 3 units to the left and 2 units up, what are the coordinates of the image vertices?
A triangle has vertices at (-2, 1), (-4, 3), and (-6, 5). If the triangle is translated 2 units to the right and 1 unit down, what are the coordinates of the image vertices?

For the first problem, the translation vector is (-3, 2). Adding this vector to the original coordinates of each vertex, we get:
- (2 - 3, 3 + 2) = (-1, 5)
- (4 - 3, 5 + 2) = (1, 7)
- (6 - 3, 7 + 2) = (3, 9)
For the second problem, the translation vector is (2, -1). Adding this vector to the original coordinates of each vertex, we get:
- (-2 + 2, 1 - 1) = (0, 0)
- (-4 + 2, 3 - 1) = (-2, 2)
- (-6 + 2, 5 - 1) = (-4, 4)

A triangle has vertices at (1, 2), (3, 4), and (5, 6). If the triangle is translated 2 units to the left and 1 unit up, what are the coordinates of the image vertices?
A triangle has vertices at (-1, -2), (-3, -4), and (-5, -6). If the triangle is translated 3 units to the right and 2 units down, what are the coordinates of the image vertices?

For the first problem, the translation vector is (-2, 1). Adding this vector to the original coordinates of each vertex, we get:
- (1 - 2, 2 + 1) = (-1, 3)
- (3 - 2, 4 + 1) = (1, 5)
- (5 - 2, 6 + 1) = (3, 7)
For the second problem, the translation vector is (3, -2). Adding this vector to the original coordinates of each vertex, we get:
- (-1 + 3, -2 - 2) = (2, -4)
- (-3 + 3, -4 - 2) = (0, -6)
- (-5 + 3, -6 - 2) = (-2, -8)