Translation of Points in a Coordinate Plane

Oct 13, 2025 by windailys.com 11 views

Introduction

In mathematics, the concept of translation is a fundamental aspect of geometry, where a point or a shape is moved from one location to another without any change in its size or orientation. In this article, we will explore the translation of a triangle in a coordinate plane, focusing on the movement of one of its vertices. We will use the given vertices of triangle ABC and apply the translation to find the new coordinates of vertex B'.

Translation of Points in a Coordinate Plane

To understand the translation of points in a coordinate plane, let's consider the basic concept of movement. When a point is translated, it moves from its original position to a new position, and the distance between the two points remains the same. In a coordinate plane, the translation of a point can be represented by changing its x-coordinate and/or y-coordinate.

Translation of Triangle ABC

Given the vertices of triangle ABC as A(4,5), B(0,5), and C(3,-1), we are asked to translate the triangle 3 units to the left and 1 unit up. This means that each vertex of the triangle will move 3 units to the left and 1 unit up.

Translation of Vertex B

To find the new coordinates of vertex B' after the translation, we need to apply the translation to the original coordinates of vertex B. The original coordinates of vertex B are (0,5). Since the translation is 3 units to the left and 1 unit up, we need to subtract 3 from the x-coordinate and add 1 to the y-coordinate.

Calculating the New Coordinates of Vertex B'

Let's calculate the new coordinates of vertex B' by applying the translation to the original coordinates of vertex B.

x-coordinate: 0 - 3 = -3
y-coordinate: 5 + 1 = 6

Therefore, the new coordinates of vertex B' are (-3,6).

Conclusion

In this article, we explored the translation of a triangle in a coordinate plane, focusing on the movement of one of its vertices. We applied the translation to the original coordinates of vertex B and calculated the new coordinates of vertex B'. The new coordinates of vertex B' are (-3,6). This example demonstrates the concept of translation in a coordinate plane and how it can be applied to find the new coordinates of points after movement.

Example Problems

Given the vertices of triangle DEF as D(2,3), E(5,3), and F(4,1), translate the triangle 2 units to the right and 2 units up. Find the new coordinates of vertex E'.
Given the vertices of triangle GHI as G(1,2), H(4,2), and I(3,0), translate the triangle 1 unit to the left and 3 units up. Find the new coordinates of vertex H'.

Solutions

To find the new coordinates of vertex E', we need to apply the translation to the original coordinates of vertex E. The original coordinates of vertex E are (5,3). Since the translation is 2 units to the right and 2 units up, we need to add 2 to the x-coordinate and add 2 to the y-coordinate.
- x-coordinate: 5 + 2 = 7
- y-coordinate: 3 + 2 = 5
Therefore, the new coordinates of vertex E' are (7,5).
To find the new coordinates of vertex H', we need to apply the translation to the original coordinates of vertex H. The original coordinates of vertex H are (4,2). Since the translation is 1 unit to the left and 3 units up, we need to subtract 1 from the x-coordinate and add 3 to the y-coordinate.
- x-coordinate: 4 - 1 = 3
- y-coordinate: 2 + 3 = 5
Therefore, the new coordinates of vertex H' are (3,5).

Practice Problems

Given the vertices of triangle JKL as J(3,4), K(6,4), and L(5,1), translate the triangle 1 unit to the left and 2 units down. Find the new coordinates of vertex K'.
Given the vertices of triangle MNO as M(2,1), N(5,1), and O(4,0), translate the triangle 2 units to the right and 3 units up. Find the new coordinates of vertex N'.

Answer Key

The new coordinates of vertex K' are (5,2).
The new coordinates of vertex N' are (7,4).

Glossary

Translation: The movement of a point or a shape from one location to another without any change in its size or orientation.
Coordinate Plane: A two-dimensional plane with a set of points represented by their x-coordinates and y-coordinates.
Vertex: A point where two or more edges of a shape meet.

References

[1] "Geometry: A Comprehensive Introduction". By Dan Pedoe. Published by Dover Publications.
[2] "Mathematics for Elementary Teachers: A Conceptual Approach". By Gary L. Musser and Judah L. Schwartz. Published by John Wiley & Sons.

About the Author

Q: What is the difference between translation and rotation in geometry?

A: Translation and rotation are two fundamental concepts in geometry. Translation is the movement of a point or a shape from one location to another without any change in its size or orientation. Rotation, on the other hand, is the movement of a point or a shape around a fixed point or axis, resulting in a change in its orientation.

Q: How do you translate a point in a coordinate plane?

A: To translate a point in a coordinate plane, you need to change its x-coordinate and/or y-coordinate. If the translation is to the right, you add a value to the x-coordinate. If the translation is to the left, you subtract a value from the x-coordinate. If the translation is up, you add a value to the y-coordinate. If the translation is down, you subtract a value from the y-coordinate.

Q: What is the formula for translating a point in a coordinate plane?

A: The formula for translating a point (x, y) by a distance (h, k) is:

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

where (x', y') are the new coordinates of the point after translation.

Q: Can you give an example of translating a triangle in a coordinate plane?

A: Let's consider a triangle with vertices A(2, 3), B(5, 3), and C(4, 1). If we translate the triangle 2 units to the right and 1 unit up, the new coordinates of the vertices will be:

A'(4, 4) B'(7, 4) C'(6, 2)

Q: How do you find the new coordinates of a vertex after translation?

A: To find the new coordinates of a vertex after translation, you need to apply the translation to the original coordinates of the vertex. If the translation is to the right, you add a value to the x-coordinate. If the translation is to the left, you subtract a value from the x-coordinate. If the translation is up, you add a value to the y-coordinate. If the translation is down, you subtract a value from the y-coordinate.

Q: Can you give an example of finding the new coordinates of a vertex after translation?

A: Let's consider a triangle with vertices A(2, 3), B(5, 3), and C(4, 1). If we translate the triangle 2 units to the right and 1 unit up, the new coordinates of vertex B will be:

B'(7, 4)

Q: What is the difference between translation and reflection in geometry?

A: Translation and reflection are two fundamental concepts in geometry. Translation is the movement of a point or a shape from one location to another without any change in its size or orientation. Reflection, on the other hand, is the movement of a point or a shape across a line or a plane, resulting in a change in its orientation.

Q: Can you give an example of reflecting a triangle in a coordinate plane?

A: Let's consider a triangle with vertices A(2, 3), B(5, 3), and C(4, 1). If we reflect the triangle across the x-axis, the new coordinates of the vertices will be:

A'(2, -3) B'(5, -3) C'(4, -1)

Q: How do you find the new coordinates of a vertex after reflection?

A: To find the new coordinates of a vertex after reflection, you need to apply the reflection to the original coordinates of the vertex. If the reflection is across the x-axis, you change the sign of the y-coordinate. If the reflection is across the y-axis, you change the sign of the x-coordinate.

Q: Can you give an example of finding the new coordinates of a vertex after reflection?

A: Let's consider a triangle with vertices A(2, 3), B(5, 3), and C(4, 1). If we reflect the triangle across the x-axis, the new coordinates of vertex A will be:

A'(2, -3)

Conclusion

In this article, we have discussed the concept of translation in geometry, including the formula for translating a point in a coordinate plane and examples of translating triangles. We have also discussed the concept of reflection in geometry, including the formula for reflecting a point in a coordinate plane and examples of reflecting triangles. We hope that this article has provided a clear understanding of these fundamental concepts in geometry.