A Pentagon Is Transformed According To The Rule R 0 , 180 ∘ R_{0,180^{\circ}} R 0 , 18 0 ∘ ​ . Which Is Another Way To State The Transformation?A. { (x, Y) \rightarrow (-x, -y)$}$B. { (x, Y) \rightarrow (-y, -x)$} C . \[ C. \[ C . \[ (x, Y) \rightarrow (x,

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Introduction

In geometry, transformations play a crucial role in understanding the properties and behavior of shapes. One of the fundamental transformations is the rotation, which involves rotating a shape around a fixed point by a certain angle. In this article, we will explore the transformation rule $R_{0,180^{\circ}}$ and its equivalent representation.

Understanding the Transformation Rule $R_{0,180^{\circ}}$

The transformation rule $R_{0,180^{\circ}}$ involves rotating a shape by $180^{\circ}$ around the origin (0, 0). This means that every point on the shape will be moved to a new location that is $180^{\circ}$ counterclockwise from its original position.

Representing the Transformation Rule

To represent the transformation rule $R_{0,180^{\circ}}$ mathematically, we can use the following notation:

$$R_{0,180^{\circ}}: (x, y) \rightarrow (x', y')$$

where $(x, y)$ is the original point and $(x', y')$ is the new point after the transformation.

Finding the Equivalent Representation

To find the equivalent representation of the transformation rule $R_{0,180^{\circ}}$, we need to determine the new coordinates $(x', y')$ in terms of the original coordinates $(x, y)$.

Using the rotation formula, we can write:

$$x' = x \cos(180^{\circ}) - y \sin(180^{\circ})$$

$$y' = x \sin(180^{\circ}) + y \cos(180^{\circ})$$

Since $\cos(180^{\circ}) = -1$ and $\sin(180^{\circ}) = 0$, we can simplify the equations to:

$$x' = -x$$

$$y' = -y$$

Therefore, the equivalent representation of the transformation rule $R_{0,180^{\circ}}$ is:

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

Comparing with the Options

Now that we have found the equivalent representation of the transformation rule $R_{0,180^{\circ}}$, let's compare it with the options provided:

A. $(x, y) \rightarrow (-x, -y)$ B. $(x, y) \rightarrow (-y, -x)$ C. $(x, y) \rightarrow (x, y)$

We can see that option A matches our equivalent representation, while options B and C do not.

Conclusion

In conclusion, the transformation rule $R_{0,180^{\circ}}$ can be represented as $(x, y) \rightarrow (-x, -y)$. This equivalent representation helps us understand the behavior of the transformation and how it affects the coordinates of a shape.

Key Takeaways

• The transformation rule $R_{0,180^{\circ}}$ involves rotating a shape by $180^{\circ}$ around the origin.
• The equivalent representation of the transformation rule is $(x, y) \rightarrow (-x, -y)$.
• This representation helps us understand the behavior of the transformation and how it affects the coordinates of a shape.

Further Reading

For more information on transformations and geometry, you can refer to the following resources:

• Geometric Transformations
• Rotation in Geometry
• Coordinate Geometry

Practice Problems

Try the following practice problems to test your understanding of the transformation rule $R_{0,180^{\circ}}$:

• Rotate the point $(3, 4)$ by $180^{\circ}$ around the origin.
• Find the equivalent representation of the transformation rule $R_{0,180^{\circ}}$.

Glossary

• Transformation: A change in the position, size, or orientation of a shape.
• Rotation: A transformation that involves rotating a shape around a fixed point by a certain angle.
• Coordinate Geometry: The study of geometric shapes using coordinates and transformations.

References

• Geometry
• Transformations
• Coordinate Geometry
  

Introduction

In our previous article, we explored the transformation rule $R_{0,180^{\circ}}$ and its equivalent representation. In this article, we will answer some frequently asked questions about the transformation rule and provide additional insights.

Q&A

Q: What is the transformation rule $R_{0,180^{\circ}}$?

A: The transformation rule $R_{0,180^{\circ}}$ involves rotating a shape by $180^{\circ}$ around the origin (0, 0).

Q: How do I represent the transformation rule $R_{0,180^{\circ}}$ mathematically?

A: You can represent the transformation rule $R_{0,180^{\circ}}$ mathematically using the following notation:

$$R_{0,180^{\circ}}: (x, y) \rightarrow (x', y')$$

where $(x, y)$ is the original point and $(x', y')$ is the new point after the transformation.

Q: What is the equivalent representation of the transformation rule $R_{0,180^{\circ}}$?

A: The equivalent representation of the transformation rule $R_{0,180^{\circ}}$ is:

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

Q: How do I apply the transformation rule $R_{0,180^{\circ}}$ to a point?

A: To apply the transformation rule $R_{0,180^{\circ}}$ to a point, you can use the following steps:

1. Identify the original point $(x, y)$.
2. Apply the transformation rule $R_{0,180^{\circ}}$ to the point by replacing $x$ with $-x$ and $y$ with $-y$.
3. The new point $(x', y')$ is the result of the transformation.

Q: What is the effect of the transformation rule $R_{0,180^{\circ}}$ on the coordinates of a shape?

A: The transformation rule $R_{0,180^{\circ}}$ has the effect of negating the coordinates of a shape. This means that the x-coordinate and y-coordinate of each point on the shape are both negated.

Q: Can I use the transformation rule $R_{0,180^{\circ}}$ to rotate a shape by a different angle?

A: No, the transformation rule $R_{0,180^{\circ}}$ is specifically designed to rotate a shape by $180^{\circ}$. If you need to rotate a shape by a different angle, you will need to use a different transformation rule.

Additional Insights

• The transformation rule $R_{0,180^{\circ}}$ is a type of rotation transformation.
• The equivalent representation of the transformation rule $R_{0,180^{\circ}}$ is $(-x, -y)$.
• The transformation rule $R_{0,180^{\circ}}$ can be applied to any shape, including points, lines, and polygons.

Practice Problems

Try the following practice problems to test your understanding of the transformation rule $R_{0,180^{\circ}}$:

• Rotate the point $(3, 4)$ by $180^{\circ}$ around the origin.
• Find the equivalent representation of the transformation rule $R_{0,180^{\circ}}$.
• Apply the transformation rule $R_{0,180^{\circ}}$ to the point $(2, 3)$.

Glossary

• Transformation: A change in the position, size, or orientation of a shape.
• Rotation: A transformation that involves rotating a shape around a fixed point by a certain angle.
• Coordinate Geometry: The study of geometric shapes using coordinates and transformations.

References

• Geometry
• Transformations
• Coordinate Geometry