A Triangle Is Rotated $90^{\circ}$ About The Origin. Which Rule Describes The Transformation?A. $(x, Y) \rightarrow (-x, -y$\] B. $(x, Y) \rightarrow (-y, X$\] C. $(x, Y) \rightarrow (-y, -x$\] D. $(x, Y)

Introduction

In geometry, transformations are essential concepts that help us understand how shapes change under various operations. One common transformation is rotation, where a shape is rotated by a certain angle around a fixed point, known as the origin. In this article, we will explore the transformation rule that describes a triangle rotated $90^{\circ}$ about the origin.

What is a Rotation?

A rotation is a transformation that turns a shape around a fixed point, called the origin, by a certain angle. In this case, we are dealing with a $90^{\circ}$ rotation, which means the triangle will be rotated by $90$ degrees around the origin.

Understanding the Transformation Rule

To describe the transformation rule, we need to understand how the coordinates of the triangle's vertices change after the rotation. Let's consider a point $(x, y)$ on the triangle. After a $90^{\circ}$ rotation about the origin, the new coordinates of the point will be $(x', y')$. We need to find the relationship between $(x, y)$ and $(x', y')$.

Analyzing the Options

Let's analyze the given options to determine which one describes the transformation rule:

Option A: $(x, y) \rightarrow (-x, -y)$

This option suggests that the x-coordinate becomes negative, and the y-coordinate also becomes negative. However, this is not a $90^{\circ}$ rotation, as the x and y coordinates are not swapped.

Option B: $(x, y) \rightarrow (-y, x)$

This option suggests that the x-coordinate becomes the y-coordinate, and the y-coordinate becomes the negative of the original x-coordinate. This is a $90^{\circ}$ rotation, as the x and y coordinates are swapped, and the sign of one coordinate is changed.

Option C: $(x, y) \rightarrow (-y, -x)$

This option suggests that both the x and y coordinates become negative, and the coordinates are swapped. However, this is not a $90^{\circ}$ rotation, as the sign of both coordinates is changed.

Option D: $(x, y) \rightarrow (y, -x)$

This option suggests that the x-coordinate becomes the y-coordinate, and the y-coordinate becomes the negative of the original x-coordinate. However, this is not a $90^{\circ}$ rotation, as the sign of the y-coordinate is changed.

Conclusion

Based on our analysis, we can conclude that the transformation rule that describes a triangle rotated $90^{\circ}$ about the origin is:

$(x, y) \rightarrow (-y, x)$

This rule indicates that the x-coordinate becomes the y-coordinate, and the y-coordinate becomes the negative of the original x-coordinate. This is a $90^{\circ}$ rotation, as the x and y coordinates are swapped, and the sign of one coordinate is changed.

Example

Let's consider an example to illustrate the transformation rule. Suppose we have a triangle with vertices at $(2, 3)$, $(4, 5)$, and $(6, 7)$. After a $90^{\circ}$ rotation about the origin, the new coordinates of the vertices will be:

• $(2, 3) \rightarrow (-3, 2)$
• $(4, 5) \rightarrow (-5, 4)$
• $(6, 7) \rightarrow (-7, 6)$

As we can see, the x and y coordinates are swapped, and the sign of one coordinate is changed, which is consistent with the transformation rule.

Applications

The transformation rule for a $90^{\circ}$ rotation about the origin has numerous applications in mathematics, physics, and engineering. For example, it is used to describe the motion of objects in two-dimensional space, such as the rotation of a wheel or the movement of a pendulum. It is also used in computer graphics to create animations and simulations.

Conclusion

Q&A: Frequently Asked Questions

Q: What is a rotation in geometry?

A: A rotation is a transformation that turns a shape around a fixed point, called the origin, by a certain angle. In this case, we are dealing with a $90^{\circ}$ rotation, which means the triangle will be rotated by $90$ degrees around the origin.

Q: How do I determine the transformation rule for a $90^{\circ}$ rotation?

A: To determine the transformation rule, you need to understand how the coordinates of the triangle's vertices change after the rotation. Let's consider a point $(x, y)$ on the triangle. After a $90^{\circ}$ rotation about the origin, the new coordinates of the point will be $(x', y')$. You need to find the relationship between $(x, y)$ and $(x', y')$.

Q: What is the transformation rule for a $90^{\circ}$ rotation about the origin?

A: The transformation rule for a $90^{\circ}$ rotation about the origin is $(x, y) \rightarrow (-y, x)$. This rule indicates that the x-coordinate becomes the y-coordinate, and the y-coordinate becomes the negative of the original x-coordinate.

Q: Can you provide an example to illustrate the transformation rule?

A: Let's consider an example to illustrate the transformation rule. Suppose we have a triangle with vertices at $(2, 3)$, $(4, 5)$, and $(6, 7)$. After a $90^{\circ}$ rotation about the origin, the new coordinates of the vertices will be:

• $(2, 3) \rightarrow (-3, 2)$
• $(4, 5) \rightarrow (-5, 4)$
• $(6, 7) \rightarrow (-7, 6)$

As we can see, the x and y coordinates are swapped, and the sign of one coordinate is changed, which is consistent with the transformation rule.

Q: What are some applications of the transformation rule for a $90^{\circ}$ rotation?

A: The transformation rule for a $90^{\circ}$ rotation about the origin has numerous applications in mathematics, physics, and engineering. For example, it is used to describe the motion of objects in two-dimensional space, such as the rotation of a wheel or the movement of a pendulum. It is also used in computer graphics to create animations and simulations.

Q: Can I use the transformation rule for a $90^{\circ}$ rotation to solve problems in geometry?

A: Yes, you can use the transformation rule for a $90^{\circ}$ rotation to solve problems in geometry. For example, you can use it to find the coordinates of a point after a $90^{\circ}$ rotation, or to determine the shape of a figure after a $90^{\circ}$ rotation.

Q: How do I apply the transformation rule for a $90^{\circ}$ rotation in a real-world scenario?

A: To apply the transformation rule for a $90^{\circ}$ rotation in a real-world scenario, you need to understand the problem and the transformation that is being applied. You can then use the transformation rule to find the new coordinates of the object or shape after the rotation.

Conclusion

In conclusion, the transformation rule for a $90^{\circ}$ rotation about the origin is $(x, y) \rightarrow (-y, x)$. This rule indicates that the x-coordinate becomes the y-coordinate, and the y-coordinate becomes the negative of the original x-coordinate. The transformation rule has numerous applications in mathematics, physics, and engineering, and is an essential concept in geometry and computer graphics.