Calculating the Image Coordinates of Triangle EFG

Oct 13, 2025 by windailys.com 98 views

Introduction

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 focuses on the translation of a triangle, specifically the triangle EFG, and explores the process of calculating the image coordinates of the translated triangle. We will delve into the concept of translation, understand the transformation process, and apply it to the given triangle EFG.

What is Translation?

Translation is a type of transformation that involves moving a figure from one location to another without changing its size or shape. It is a rigid motion that preserves the distance and angle between the points of the figure. In other words, translation is a movement of a figure in a specific direction and distance without any rotation or scaling.

Types of Translation

There are two types of translation:

Horizontal Translation: This type of translation involves moving a figure horizontally, i.e., left or right.
Vertical Translation: This type of translation involves moving a figure vertically, i.e., up or down.

Translation Formula

The translation formula is used to calculate the image coordinates of a translated figure. The formula is given by:

(x', y') = (x + h, y + k)

where (x, y) are the coordinates of the original figure, and (h, k) are the translation values.

Calculating the Image Coordinates of Triangle EFG

The 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' (-1, 0), F' (-3, -5), and G' (3, -3).

To calculate the image coordinates of the translated triangle, we need to apply the translation formula to each vertex of the triangle.

Calculating the Image Coordinates of Vertex E

The coordinates of vertex E are (-3, 4). The translation values are (-1, 0). Applying the translation formula, we get:

x' = x + h = -3 + (-1) = -4 y' = y + k = 4 + 0 = 4

Therefore, the image coordinates of vertex E' are (-4, 4).

Calculating the Image Coordinates of Vertex F

The coordinates of vertex F are (-5, -1). The translation values are (-1, 0). Applying the translation formula, we get:

x' = x + h = -5 + (-1) = -6 y' = y + k = -1 + 0 = -1

Therefore, the image coordinates of vertex F' are (-6, -1).

Calculating the Image Coordinates of Vertex G

The coordinates of vertex G are (1, 1). The translation values are (3, -3). Applying the translation formula, we get:

x' = x + h = 1 + 3 = 4 y' = y + k = 1 + (-3) = -2

Therefore, the image coordinates of vertex G' are (4, -2).

Conclusion

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. The translation formula is used to calculate the image coordinates of a translated figure. By applying the translation formula to each vertex of the triangle EFG, we can calculate the image coordinates of the translated triangle. The image coordinates of the translated triangle EFG are E' (-4, 4), F' (-6, -1), and G' (4, -2).

Example Problems

A triangle has vertices A (2, 3), B (4, 5), and C (6, 7). The triangle is translated so that the coordinates of the image are A' (1, 2), B' (3, 4), and C' (5, 6). Calculate the translation values.
A square has vertices D (1, 1), E (3, 1), F (3, 3), and G (1, 3). The square is translated so that the coordinates of the image are D' (2, 2), E' (4, 2), F' (4, 4), and G' (2, 4). Calculate the translation values.

Solutions

To calculate the translation values, we need to apply the translation formula to each vertex of the triangle.

Applying the translation formula to vertex A, we get:

x' = x + h = 2 + 1 = 3 y' = y + k = 3 + 2 = 5

Therefore, the image coordinates of vertex A' are (3, 5).

Applying the translation formula to vertex B, we get:

x' = x + h = 4 + 1 = 5 y' = y + k = 5 + 2 = 7

Therefore, the image coordinates of vertex B' are (5, 7).

Applying the translation formula to vertex C, we get:

x' = x + h = 6 + 1 = 7 y' = y + k = 7 + 2 = 9

Therefore, the image coordinates of vertex C' are (7, 9).

The translation values are (1, 2).
To calculate the translation values, we need to apply the translation formula to each vertex of the square.

Applying the translation formula to vertex D, we get:

x' = x + h = 1 + 2 = 3 y' = y + k = 1 + 2 = 3

Therefore, the image coordinates of vertex D' are (3, 3).

Applying the translation formula to vertex E, we get:

x' = x + h = 3 + 2 = 5 y' = y + k = 1 + 2 = 3

Therefore, the image coordinates of vertex E' are (5, 3).

Applying the translation formula to vertex F, we get:

x' = x + h = 3 + 2 = 5 y' = y + k = 3 + 2 = 5

Therefore, the image coordinates of vertex F' are (5, 5).

Applying the translation formula to vertex G, we get:

x' = x + h = 1 + 2 = 3 y' = y + k = 3 + 2 = 5

Therefore, the image coordinates of vertex G' are (3, 5).

The translation values are (2, 2).

Practice Problems

A triangle has vertices H (2, 4), I (4, 6), and J (6, 8). The triangle is translated so that the coordinates of the image are H' (1, 3), I' (3, 5), and J' (5, 7). Calculate the translation values.
A square has vertices K (1, 1), L (3, 1), M (3, 3), and N (1, 3). The square is translated so that the coordinates of the image are K' (2, 2), L' (4, 2), M' (4, 4), and N' (2, 4). Calculate the translation values.

Answers

The translation values are (1, 1).
The translation values are (2, 2).

Conclusion

Q: What is translation in geometry?

A: Translation is a type of transformation that involves moving a figure from one location to another without changing its size or shape. It is a rigid motion that preserves the distance and angle between the points of the figure.

Q: What are the two types of translation?

A: There are two types of translation:

Horizontal Translation: This type of translation involves moving a figure horizontally, i.e., left or right.
Vertical Translation: This type of translation involves moving a figure vertically, i.e., up or down.

Q: What is the translation formula?

A: The translation formula is used to calculate the image coordinates of a translated figure. The formula is given by:

(x', y') = (x + h, y + k)

where (x, y) are the coordinates of the original figure, and (h, k) are the translation values.

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

A: To calculate the image coordinates of a translated triangle, you need to apply the translation formula to each vertex of the triangle.

Q: What are the image coordinates of the translated triangle EFG?

A: The image coordinates of the translated triangle EFG are E' (-4, 4), F' (-6, -1), and G' (4, -2).

Q: How do I calculate the translation values of a translated square?

A: To calculate the translation values of a translated square, you need to apply the translation formula to each vertex of the square.

Q: What are the translation values of the translated square KLMN?

A: The translation values of the translated square KLMN are (2, 2).

Q: Can I use the translation formula to calculate the image coordinates of a translated figure with more than three vertices?

A: Yes, you can use the translation formula to calculate the image coordinates of a translated figure with more than three vertices.

Q: How do I apply the translation formula to a translated figure with more than three vertices?

A: To apply the translation formula to a translated figure with more than three vertices, you need to apply the formula to each vertex of the figure.

Q: What are the image coordinates of the translated figure with more than three vertices?

A: The image coordinates of the translated figure with more than three vertices are calculated by applying the translation formula to each vertex of the figure.

Q: Can I use the translation formula to calculate the image coordinates of a translated figure with negative coordinates?

A: Yes, you can use the translation formula to calculate the image coordinates of a translated figure with negative coordinates.

Q: How do I apply the translation formula to a translated figure with negative coordinates?

A: To apply the translation formula to a translated figure with negative coordinates, you need to apply the formula to each vertex of the figure.

Q: What are the image coordinates of the translated figure with negative coordinates?

A: The image coordinates of the translated figure with negative coordinates are calculated by applying the translation formula to each vertex of the figure.

Q: Can I use the translation formula to calculate the image coordinates of a translated figure with decimal coordinates?

A: Yes, you can use the translation formula to calculate the image coordinates of a translated figure with decimal coordinates.

Q: How do I apply the translation formula to a translated figure with decimal coordinates?

A: To apply the translation formula to a translated figure with decimal coordinates, you need to apply the formula to each vertex of the figure.

Q: What are the image coordinates of the translated figure with decimal coordinates?

A: The image coordinates of the translated figure with decimal coordinates are calculated by applying the translation formula to each vertex of the figure.