What Is The Mapping Rule For A 180-degree Rotation About The Origin?A. \[$(x, Y) \rightarrow (-y, X)\$\]B. \[$(x, Y) \rightarrow (-y, -x)\$\]C. \[$(x, Y) \rightarrow (-x, -y)\$\]D. \[$(x, Y) \rightarrow (x, -y)\$\]

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Understanding Rotations in Mathematics

Rotations are fundamental concepts in mathematics, particularly in geometry and trigonometry. A rotation is a transformation that turns a figure around a fixed point called the origin. In this article, we will focus on a 180-degree rotation about the origin, which is a crucial concept in mathematics.

What is a 180-Degree Rotation?

A 180-degree rotation is a transformation that turns a figure around the origin by 180 degrees. This means that the figure is rotated completely upside down. To understand the mapping rule for a 180-degree rotation, we need to consider how the coordinates of a point change after the rotation.

Mapping Rule for a 180-Degree Rotation

The mapping rule for a 180-degree rotation about the origin is given by the formula:

(x,y)→(−y,−x) (x, y) \rightarrow (-y, -x)

This formula indicates that the x-coordinate of a point becomes the negative of the y-coordinate, and the y-coordinate becomes the negative of the x-coordinate.

Explanation of the Mapping Rule

To understand why the mapping rule is given by the formula above, let's consider a point (x, y) on the coordinate plane. When we rotate this point by 180 degrees about the origin, the new coordinates of the point are given by (-y, -x). This is because the x-coordinate of the point is now the negative of the y-coordinate, and the y-coordinate is now the negative of the x-coordinate.

Example of a 180-Degree Rotation

Let's consider an example to illustrate the mapping rule for a 180-degree rotation. Suppose we have a point (3, 4) on the coordinate plane. When we rotate this point by 180 degrees about the origin, the new coordinates of the point are given by (-4, -3).

Comparison with Other Options

Now, let's compare the mapping rule for a 180-degree rotation with the other options given in the problem.

  • Option A: (x,y)→(−y,x)(x, y) \rightarrow (-y, x) This option is incorrect because it does not give the correct mapping rule for a 180-degree rotation.
  • Option B: (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x) This option is correct because it gives the same mapping rule as the formula above.
  • Option C: (x,y)→(−x,−y)(x, y) \rightarrow (-x, -y) This option is incorrect because it does not give the correct mapping rule for a 180-degree rotation.
  • Option D: (x,y)→(x,−y)(x, y) \rightarrow (x, -y) This option is incorrect because it does not give the correct mapping rule for a 180-degree rotation.

Conclusion

In conclusion, the mapping rule for a 180-degree rotation about the origin is given by the formula:

(x,y)→(−y,−x) (x, y) \rightarrow (-y, -x)

This formula indicates that the x-coordinate of a point becomes the negative of the y-coordinate, and the y-coordinate becomes the negative of the x-coordinate. This is a crucial concept in mathematics, particularly in geometry and trigonometry.

Frequently Asked Questions

  • What is a 180-degree rotation? A 180-degree rotation is a transformation that turns a figure around the origin by 180 degrees.
  • What is the mapping rule for a 180-degree rotation? The mapping rule for a 180-degree rotation is given by the formula: (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x).
  • Why is the mapping rule given by the formula above? The mapping rule is given by the formula above because it indicates that the x-coordinate of a point becomes the negative of the y-coordinate, and the y-coordinate becomes the negative of the x-coordinate.

Final Thoughts

In conclusion, the mapping rule for a 180-degree rotation about the origin is a crucial concept in mathematics. Understanding this concept is essential for solving problems in geometry and trigonometry. We hope that this article has provided a clear explanation of the mapping rule for a 180-degree rotation and has helped you to understand this concept better.

Understanding Rotations in Mathematics

Rotations are fundamental concepts in mathematics, particularly in geometry and trigonometry. A rotation is a transformation that turns a figure around a fixed point called the origin. In this article, we will focus on a 180-degree rotation about the origin, which is a crucial concept in mathematics.

Q&A: 180-Degree Rotation

Q1: What is a 180-degree rotation?

A1: A 180-degree rotation is a transformation that turns a figure around the origin by 180 degrees. This means that the figure is rotated completely upside down.

Q2: What is the mapping rule for a 180-degree rotation?

A2: The mapping rule for a 180-degree rotation is given by the formula: (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x). This formula indicates that the x-coordinate of a point becomes the negative of the y-coordinate, and the y-coordinate becomes the negative of the x-coordinate.

Q3: Why is the mapping rule given by the formula above?

A3: The mapping rule is given by the formula above because it indicates that the x-coordinate of a point becomes the negative of the y-coordinate, and the y-coordinate becomes the negative of the x-coordinate.

Q4: What happens to the coordinates of a point when it is rotated by 180 degrees?

A4: When a point is rotated by 180 degrees, its coordinates are swapped and made negative. For example, if a point has coordinates (3, 4), its coordinates after a 180-degree rotation would be (-4, -3).

Q5: How do I apply the mapping rule for a 180-degree rotation to a point?

A5: To apply the mapping rule for a 180-degree rotation to a point, you need to swap the coordinates of the point and make them negative. For example, if a point has coordinates (3, 4), you would swap the coordinates to get (4, 3) and then make them negative to get (-4, -3).

Q6: What is the difference between a 180-degree rotation and a 90-degree rotation?

A6: A 180-degree rotation is a transformation that turns a figure around the origin by 180 degrees, while a 90-degree rotation is a transformation that turns a figure around the origin by 90 degrees. The mapping rule for a 180-degree rotation is given by the formula: (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x), while the mapping rule for a 90-degree rotation is given by the formula: (x,y)→(−y,x)(x, y) \rightarrow (-y, x).

Q7: Can I apply the mapping rule for a 180-degree rotation to a point in 3D space?

A7: Yes, you can apply the mapping rule for a 180-degree rotation to a point in 3D space. The mapping rule would be given by the formula: (x,y,z)→(−y,−x,−z)(x, y, z) \rightarrow (-y, -x, -z).

Q8: How do I find the image of a point after a 180-degree rotation?

A8: To find the image of a point after a 180-degree rotation, you need to apply the mapping rule for a 180-degree rotation to the point. For example, if a point has coordinates (3, 4), you would swap the coordinates to get (4, 3) and then make them negative to get (-4, -3).

Q9: Can I apply the mapping rule for a 180-degree rotation to a line or a plane?

A9: Yes, you can apply the mapping rule for a 180-degree rotation to a line or a plane. The mapping rule would be given by the formula: (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x) for a line or plane in 2D space, and (x,y,z)→(−y,−x,−z)(x, y, z) \rightarrow (-y, -x, -z) for a line or plane in 3D space.

Q10: What is the importance of understanding 180-degree rotation in mathematics?

A10: Understanding 180-degree rotation is important in mathematics because it is a fundamental concept in geometry and trigonometry. It is used to solve problems in these fields, and it is also used in many real-world applications, such as computer graphics and engineering.

Conclusion

In conclusion, the mapping rule for a 180-degree rotation is a crucial concept in mathematics. Understanding this concept is essential for solving problems in geometry and trigonometry. We hope that this article has provided a clear explanation of the mapping rule for a 180-degree rotation and has helped you to understand this concept better.