What Is The Coordinate For The Image Of Point $H(2, -6)$ Under A $90^{\circ}$ Clockwise Rotation About The Origin?A. $H(-6, -2$\] B. $H^{\prime}(6, -2$\] C. $H^{\prime}(-6, 2$\] D. $H^{\prime}(6,
Introduction
In geometry, a rotation is a transformation that turns a figure around a fixed point called the center of rotation. When a point is rotated about the origin, its coordinates change according to specific rules. In this article, we will explore the concept of a $90^{\circ}$ clockwise rotation about the origin and determine the new coordinates of a point $H(2, -6)$ after such a transformation.
Understanding Rotations
A rotation is a type of rigid motion that preserves the distance between points. When a point is rotated about the origin, its coordinates change in a predictable way. To understand how the coordinates change, let's consider the effect of a $90^{\circ}$ clockwise rotation on the x and y axes.
Effect on the X-Axis
When a point is rotated $90^{\circ}$ clockwise about the origin, its x-coordinate becomes the negative of its original y-coordinate. This is because the x-axis is rotated to the y-axis, and the new x-coordinate is the negative of the original y-coordinate.
Effect on the Y-Axis
Similarly, when a point is rotated $90^{\circ}$ clockwise about the origin, its y-coordinate becomes the positive of its original x-coordinate. This is because the y-axis is rotated to the x-axis, and the new y-coordinate is the positive of the original x-coordinate.
Applying the Rotation to Point $H(2, -6)$
Now that we understand how the coordinates change under a $90^{\circ}$ clockwise rotation, let's apply this transformation to point $H(2, -6)$.
New X-Coordinate
The new x-coordinate is the negative of the original y-coordinate, which is $-(-6) = 6$.
New Y-Coordinate
The new y-coordinate is the positive of the original x-coordinate, which is $2$.
Conclusion
After applying the $90^{\circ}$ clockwise rotation about the origin, the new coordinates of point $H(2, -6)$ are $H^{\prime}(6, 2)$. This means that the correct answer is:
C.
However, this is incorrect. The correct answer is actually D. . The negative sign in the original answer is incorrect, and the correct answer is the one with the positive sign.
Final Answer
The final answer is D. .
Introduction
In our previous article, we explored the concept of a $90^{\circ}$ clockwise rotation about the origin and determined the new coordinates of a point $H(2, -6)$ after such a transformation. In this article, we will answer some frequently asked questions about rotations in geometry.
Q: What is a rotation in geometry?
A: A rotation is a type of rigid motion that turns a figure around a fixed point called the center of rotation. When a point is rotated about the origin, its coordinates change according to specific rules.
Q: What is the effect of a $90^{\circ}$ clockwise rotation on the x and y axes?
A: When a point is rotated $90^{\circ}$ clockwise about the origin, its x-coordinate becomes the negative of its original y-coordinate, and its y-coordinate becomes the positive of its original x-coordinate.
Q: How do I determine the new coordinates of a point after a rotation?
A: To determine the new coordinates of a point after a rotation, you need to apply the rotation rules to the original coordinates. For a $90^{\circ}$ clockwise rotation, the new x-coordinate is the negative of the original y-coordinate, and the new y-coordinate is the positive of the original x-coordinate.
Q: What is the center of rotation?
A: The center of rotation is the fixed point around which the figure is turned. In the case of a $90^{\circ}$ clockwise rotation about the origin, the center of rotation is the origin (0, 0).
Q: Can I rotate a figure by more than $90^{\circ}$?
A: Yes, you can rotate a figure by more than $90^{\circ}$. However, the rotation rules will change depending on the angle of rotation. For example, a $180^{\circ}$ rotation will swap the x and y coordinates, while a $270^{\circ}$ rotation will swap the x and y coordinates and change the sign of the new x-coordinate.
Q: How do I visualize a rotation?
A: To visualize a rotation, you can use a coordinate grid or a graphing calculator to plot the original figure and the rotated figure. You can also use geometric software or online tools to visualize the rotation.
Q: What are some common applications of rotations in geometry?
A: Rotations have many applications in geometry, including:
- Symmetry: Rotations can be used to create symmetrical figures.
- Reflections: Rotations can be used to create reflections of figures.
- Translations: Rotations can be used to create translations of figures.
- Compositions: Rotations can be used to create compositions of figures.
Conclusion
Rotations are an important concept in geometry, and understanding how to apply them can help you solve a wide range of problems. By following the rotation rules and visualizing the rotation, you can determine the new coordinates of a point after a rotation. Whether you're working with a $90^{\circ}$ clockwise rotation or a more complex rotation, the principles of rotation remain the same.
Final Answer
The final answer is that rotations are a fundamental concept in geometry, and understanding how to apply them can help you solve a wide range of problems.