Rotating a Triangle Counterclockwise

Oct 13, 2025 by windailys.com 13 views

Introduction

In mathematics, rotation is a fundamental concept that involves turning a figure or shape around a fixed point called the origin. When a triangle is rotated counterclockwise about the origin, its vertices change positions, resulting in a new set of image points. In this article, we will explore the concept of rotating a triangle counterclockwise about the origin and determine the correct set of image points for triangle $$XYZ$$.

Understanding Rotation

Rotation is a transformation that involves turning a figure or shape around a fixed point called the origin. The origin is typically represented by the coordinates (0, 0) on a coordinate plane. When a figure is rotated counterclockwise about the origin, its vertices change positions, resulting in a new set of image points.

Rotating a Triangle Counterclockwise

To rotate a triangle counterclockwise about the origin, we can use the following formula:

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}

where (x, y) represents the original coordinates of the vertex, and (x', y') represents the new coordinates of the image point.

Applying the Rotation Formula to Triangle XYZ

Given the vertices of triangle $$XYZ$$ as $$X(1,-1)$$, $$Y(3,4)$$, and $$Z(5,-1)$$, we can apply the rotation formula to determine the correct set of image points.

Rotating Vertex X

To rotate vertex X, we substitute the coordinates (1, -1) into the rotation formula:

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ -1 \end{pmatrix}

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 \cdot 1 + (-1) \cdot (-1) \\ 1 \cdot 1 + 0 \cdot (-1) \end{pmatrix}

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}

Therefore, the image point of vertex X is (1, 1).

Rotating Vertex Y

To rotate vertex Y, we substitute the coordinates (3, 4) into the rotation formula:

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix}

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 \cdot 3 + (-1) \cdot 4 \\ 1 \cdot 3 + 0 \cdot 4 \end{pmatrix}

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} -4 \\ 3 \end{pmatrix}

Therefore, the image point of vertex Y is (-4, 3).

Rotating Vertex Z

To rotate vertex Z, we substitute the coordinates (5, -1) into the rotation formula:

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 5 \\ -1 \end{pmatrix}

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 \cdot 5 + (-1) \cdot (-1) \\ 1 \cdot 5 + 0 \cdot (-1) \end{pmatrix}

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 \\ 5 \end{pmatrix}

Therefore, the image point of vertex Z is (1, 5).

Conclusion

In conclusion, when triangle $$XYZ$$ is rotated counterclockwise about the origin, the correct set of image points is:

Vertex X: (1, 1)
Vertex Y: (-4, 3)
Vertex Z: (1, 5)

The rotation formula provides a systematic approach to determining the image points of a triangle after rotation. By applying the rotation formula to the vertices of triangle $$XYZ$$, we can determine the correct set of image points.

Applications of Rotation

Rotation is a fundamental concept in mathematics with numerous applications in various fields, including:

Computer Graphics: Rotation is used to create 3D models and animations.
Engineering: Rotation is used to design and analyze mechanical systems, such as gears and linkages.
Physics: Rotation is used to describe the motion of objects in two-dimensional space.

Final Thoughts

In this article, we explored the concept of rotating a triangle counterclockwise about the origin and determined the correct set of image points for triangle $$XYZ$$. The rotation formula provides a systematic approach to determining the image points of a triangle after rotation. By applying the rotation formula to the vertices of triangle $$XYZ$$, we can determine the correct set of image points.

Introduction

In our previous article, we explored the concept of rotating a triangle counterclockwise about the origin and determined the correct set of image points for triangle $$XYZ$$. In this article, we will answer some frequently asked questions (FAQs) about rotating a triangle counterclockwise about the origin.

Q: What is the formula for rotating a triangle counterclockwise about the origin?

A: The formula for rotating a triangle counterclockwise about the origin is:

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}

where (x, y) represents the original coordinates of the vertex, and (x', y') represents the new coordinates of the image point.

Q: How do I apply the rotation formula to a triangle?

A: To apply the rotation formula to a triangle, you need to substitute the coordinates of each vertex into the formula. For example, if you have a triangle with vertices (x1, y1), (x2, y2), and (x3, y3), you would substitute these coordinates into the formula to get the new coordinates of the image points.

Q: What is the difference between rotating a triangle counterclockwise and clockwise?

A: Rotating a triangle counterclockwise about the origin involves turning the triangle in a counterclockwise direction, while rotating a triangle clockwise involves turning the triangle in a clockwise direction. The rotation formula is the same for both counterclockwise and clockwise rotations, but the direction of rotation is different.

Q: Can I rotate a triangle more than 90 degrees?

A: Yes, you can rotate a triangle more than 90 degrees. To do this, you would need to apply the rotation formula multiple times. For example, to rotate a triangle 180 degrees, you would need to apply the rotation formula twice.

Q: How do I determine the correct set of image points for a triangle after rotation?

A: To determine the correct set of image points for a triangle after rotation, you need to apply the rotation formula to each vertex of the triangle. The new coordinates of the image points will be the result of the rotation.

Q: What are some real-world applications of rotating a triangle counterclockwise about the origin?

A: Rotating a triangle counterclockwise about the origin has numerous real-world applications, including:

Computer Graphics: Rotation is used to create 3D models and animations.
Engineering: Rotation is used to design and analyze mechanical systems, such as gears and linkages.
Physics: Rotation is used to describe the motion of objects in two-dimensional space.

Q: Can I use the rotation formula to rotate other shapes, such as rectangles or circles?

A: Yes, you can use the rotation formula to rotate other shapes, such as rectangles or circles. However, you would need to modify the formula to accommodate the specific shape you are rotating.

Conclusion

In conclusion, rotating a triangle counterclockwise about the origin is a fundamental concept in mathematics with numerous real-world applications. By understanding the rotation formula and how to apply it to a triangle, you can determine the correct set of image points for a triangle after rotation. We hope this article has answered some of your frequently asked questions about rotating a triangle counterclockwise about the origin.

Additional Resources

If you are interested in learning more about rotating a triangle counterclockwise about the origin, we recommend checking out the following resources:

Mathematics textbooks: Many mathematics textbooks cover the topic of rotation and provide examples and exercises to help you practice.
Online tutorials: There are many online tutorials and videos that provide step-by-step instructions on how to rotate a triangle counterclockwise about the origin.
Mathematics software: Many mathematics software programs, such as GeoGebra and Mathematica, allow you to rotate shapes and explore the properties of rotation.

We hope this article has been helpful in answering your questions about rotating a triangle counterclockwise about the origin. If you have any further questions, please don't hesitate to ask.