Triangle $XYZ$ Has Vertices $X(-1,-3)$, $Y(0,0)$, And $Z(1,-3)$. Malik Rotated The Triangle $90^{\circ}$ Clockwise About The Origin. What Is The Correct Set Of Image Points For Triangle
Introduction
In geometry, rotations are an essential concept that helps us understand how shapes change when they are turned around a fixed point. In this article, we will explore the impact of a 90° clockwise rotation on a given triangle. We will use the vertices of the triangle to determine the correct set of image points after the rotation.
Understanding the Rotation
A 90° clockwise rotation about the origin involves rotating the shape by 90° in a clockwise direction around the origin (0, 0). This means that the x-coordinate of each point will be replaced by its negative y-coordinate, and the y-coordinate will be replaced by its positive x-coordinate.
The Original Triangle
The original triangle XYZ has vertices X(-1, -3), Y(0, 0), and Z(1, -3). To understand the impact of the rotation, let's analyze each vertex separately.
Vertex X(-1, -3)
The x-coordinate of vertex X is -1, and the y-coordinate is -3. After a 90° clockwise rotation, the new x-coordinate will be the negative of the original y-coordinate, which is 3. The new y-coordinate will be the positive of the original x-coordinate, which is 1. Therefore, the image point of vertex X after the rotation is (3, 1).
Vertex Y(0, 0)
The x-coordinate of vertex Y is 0, and the y-coordinate is 0. After a 90° clockwise rotation, the new x-coordinate will be the negative of the original y-coordinate, which is 0. The new y-coordinate will be the positive of the original x-coordinate, which is 0. Therefore, the image point of vertex Y after the rotation is (0, 0).
Vertex Z(1, -3)
The x-coordinate of vertex Z is 1, and the y-coordinate is -3. After a 90° clockwise rotation, the new x-coordinate will be the negative of the original y-coordinate, which is 3. The new y-coordinate will be the positive of the original x-coordinate, which is 1. Therefore, the image point of vertex Z after the rotation is (3, 1).
The Correct Set of Image Points
After analyzing each vertex separately, we can conclude that the correct set of image points for triangle XYZ after a 90° clockwise rotation is:
- Vertex X(-1, -3) → (3, 1)
- Vertex Y(0, 0) → (0, 0)
- Vertex Z(1, -3) → (3, 1)
Conclusion
In this article, we explored the impact of a 90° clockwise rotation on a given triangle. We used the vertices of the triangle to determine the correct set of image points after the rotation. By understanding the concept of rotation and applying it to each vertex, we were able to find the correct image points for the triangle. This demonstrates the importance of geometry in understanding how shapes change when they are rotated or transformed in other ways.
Further Exploration
If you're interested in exploring more about geometry and rotations, here are some additional topics you might find useful:
- Reflections: Learn about the concept of reflections and how they differ from rotations.
- Translations: Understand how translations work and how they can be used to move shapes around.
- Compositions of Transformations: Learn about how to combine multiple transformations to create more complex effects.
By exploring these topics, you'll gain a deeper understanding of geometry and how shapes can be transformed in various ways.
Introduction
In our previous article, we explored the impact of a 90° clockwise rotation on a given triangle. We used the vertices of the triangle to determine the correct set of image points after the rotation. In this article, we will answer some frequently asked questions related to triangle rotation and geometry.
Q&A
Q: What is the difference between a 90° clockwise rotation and a 90° counterclockwise rotation?
A: A 90° clockwise rotation involves rotating the shape by 90° in a clockwise direction around the origin, while a 90° counterclockwise rotation involves rotating the shape by 90° in a counterclockwise direction around the origin. The main difference between the two is the direction of rotation.
Q: How do I determine the image points of a triangle after a rotation?
A: To determine the image points of a triangle after a rotation, you need to apply the rotation transformation to each vertex of the triangle. This involves replacing the x-coordinate of each point with its negative y-coordinate and replacing the y-coordinate with its positive x-coordinate.
Q: Can I rotate a triangle by any angle?
A: Yes, you can rotate a triangle by any angle. However, the rotation angle must be measured in degrees or radians, and the rotation must be performed around a fixed point, such as the origin.
Q: How do I find the image points of a triangle after a rotation if the rotation angle is not 90°?
A: To find the image points of a triangle after a rotation by an angle other than 90°, you need to use the rotation matrix formula. The rotation matrix formula is:
x' = xcos(θ) - ysin(θ) y' = xsin(θ) + ycos(θ)
where (x, y) are the original coordinates, (x', y') are the new coordinates, and θ is the rotation angle.
Q: Can I rotate a triangle by a negative angle?
A: Yes, you can rotate a triangle by a negative angle. A negative angle represents a rotation in the opposite direction of a positive angle. For example, a rotation by -90° is equivalent to a rotation by 270°.
Q: How do I determine the image points of a triangle after a rotation if the rotation is performed around a point other than the origin?
A: To determine the image points of a triangle after a rotation performed around a point other than the origin, you need to use the translation and rotation formulas. The translation formula is:
x' = x - h y' = y - k
where (x, y) are the original coordinates, (x', y') are the new coordinates, and (h, k) are the coordinates of the rotation center.
Conclusion
In this article, we answered some frequently asked questions related to triangle rotation and geometry. We covered topics such as the difference between a 90° clockwise rotation and a 90° counterclockwise rotation, how to determine the image points of a triangle after a rotation, and how to find the image points of a triangle after a rotation by an angle other than 90°.
Further Exploration
If you're interested in exploring more about geometry and rotations, here are some additional topics you might find useful:
- Reflections: Learn about the concept of reflections and how they differ from rotations.
- Translations: Understand how translations work and how they can be used to move shapes around.
- Compositions of Transformations: Learn about how to combine multiple transformations to create more complex effects.
By exploring these topics, you'll gain a deeper understanding of geometry and how shapes can be transformed in various ways.