Applying the Rotation to the Given Point

Oct 13, 2025 by miodeya.com 115 views

Introduction

Rotating points in the coordinate plane is a fundamental concept in mathematics, particularly in geometry and trigonometry. When a point is rotated about the origin, its coordinates change in a predictable manner. In this article, we will explore how to rotate points in the coordinate plane and apply this knowledge to solve a specific problem.

Understanding Rotations

A rotation is a transformation that turns a point or a figure around a fixed point, called the center of rotation. In the coordinate plane, the center of rotation is usually the origin (0, 0). When a point is rotated about the origin, its coordinates change in a way that depends on the angle of rotation.

Rotating 180° Counterclockwise

To rotate a point 180° counterclockwise about the origin, we need to follow these steps:

Reflect the point across the y-axis: When a point is reflected across the y-axis, its x-coordinate changes sign, while its y-coordinate remains the same.
Reflect the point across the x-axis: After reflecting the point across the y-axis, we need to reflect it again across the x-axis. This time, the y-coordinate changes sign, while the x-coordinate remains the same.

Applying the Rotation to the Given Point

Now, let's apply the rotation to the given point (-1, -3).

Reflect the point across the y-axis: When we reflect the point (-1, -3) across the y-axis, its x-coordinate changes sign, while its y-coordinate remains the same. The new coordinates are (1, -3).
Reflect the point across the x-axis: After reflecting the point (1, -3) across the y-axis, we need to reflect it again across the x-axis. This time, the y-coordinate changes sign, while the x-coordinate remains the same. The final coordinates are (1, 3).

Conclusion

In conclusion, when a point is rotated 180° counterclockwise about the origin, its coordinates change in a predictable manner. By following the steps outlined above, we can apply this knowledge to solve problems involving rotations in the coordinate plane.

Answer

Based on the steps outlined above, the correct answer is:

D) (1, 3)

Additional Examples

To reinforce your understanding of rotations, let's consider a few additional examples:

Rotating a point 90° counterclockwise: When a point is rotated 90° counterclockwise about the origin, its coordinates change in a way that depends on the quadrant in which the point lies.
Rotating a point 270° counterclockwise: When a point is rotated 270° counterclockwise about the origin, its coordinates change in a way that depends on the quadrant in which the point lies.

Practice Problems

To practice your understanding of rotations, try the following problems:

Rotate the point (2, 4) 180° counterclockwise about the origin.
Rotate the point (-3, 1) 90° counterclockwise about the origin.
Rotate the point (1, -2) 270° counterclockwise about the origin.

Solutions

Rotating the point (2, 4) 180° counterclockwise: The final coordinates are (-2, -4).
Rotating the point (-3, 1) 90° counterclockwise: The final coordinates are (1, -3).
Rotating the point (1, -2) 270° counterclockwise: The final coordinates are (-2, 1).

Conclusion

Q: What is the center of rotation in the coordinate plane?

A: The center of rotation in the coordinate plane is usually the origin (0, 0).

Q: What is the difference between a rotation and a reflection in the coordinate plane?

A: A rotation is a transformation that turns a point or a figure around a fixed point, called the center of rotation. A reflection is a transformation that flips a point or a figure across a line, called the axis of reflection.

Q: How do I rotate a point 180° counterclockwise about the origin?

A: To rotate a point 180° counterclockwise about the origin, you need to follow these steps:

Reflect the point across the y-axis: When a point is reflected across the y-axis, its x-coordinate changes sign, while its y-coordinate remains the same.
Reflect the point across the x-axis: After reflecting the point across the y-axis, you need to reflect it again across the x-axis. This time, the y-coordinate changes sign, while the x-coordinate remains the same.

Q: How do I rotate a point 90° counterclockwise about the origin?

A: To rotate a point 90° counterclockwise about the origin, you need to follow these steps:

Reflect the point across the y-axis: When a point is reflected across the y-axis, its x-coordinate changes sign, while its y-coordinate remains the same.
Reflect the point across the x-axis: After reflecting the point across the y-axis, you need to reflect it again across the x-axis. This time, the y-coordinate changes sign, while the x-coordinate remains the same.

Q: How do I rotate a point 270° counterclockwise about the origin?

A: To rotate a point 270° counterclockwise about the origin, you need to follow these steps:

Reflect the point across the x-axis: When a point is reflected across the x-axis, its y-coordinate changes sign, while its x-coordinate remains the same.
Reflect the point across the y-axis: After reflecting the point across the x-axis, you need to reflect it again across the y-axis. This time, the x-coordinate changes sign, while the y-coordinate remains the same.

Q: What is the effect of rotating a point 180° counterclockwise about the origin on its coordinates?

A: When a point is rotated 180° counterclockwise about the origin, its coordinates change in a way that depends on the quadrant in which the point lies. The x-coordinate changes sign, while the y-coordinate changes sign.

Q: What is the effect of rotating a point 90° counterclockwise about the origin on its coordinates?

A: When a point is rotated 90° counterclockwise about the origin, its coordinates change in a way that depends on the quadrant in which the point lies. The x-coordinate changes sign, while the y-coordinate changes sign.

Q: What is the effect of rotating a point 270° counterclockwise about the origin on its coordinates?

A: When a point is rotated 270° counterclockwise about the origin, its coordinates change in a way that depends on the quadrant in which the point lies. The x-coordinate changes sign, while the y-coordinate changes sign.

Q: How do I determine the quadrant in which a point lies in the coordinate plane?

A: To determine the quadrant in which a point lies in the coordinate plane, you need to follow these steps:

Determine the x-coordinate of the point: If the x-coordinate is positive, the point lies in the first or fourth quadrant. If the x-coordinate is negative, the point lies in the second or third quadrant.
Determine the y-coordinate of the point: If the y-coordinate is positive, the point lies in the first or second quadrant. If the y-coordinate is negative, the point lies in the third or fourth quadrant.

Q: What is the difference between a clockwise and counterclockwise rotation in the coordinate plane?

A: A clockwise rotation is a rotation that turns a point or a figure in the opposite direction of a counterclockwise rotation. In other words, a clockwise rotation is a rotation that turns a point or a figure to the right, while a counterclockwise rotation is a rotation that turns a point or a figure to the left.

Q: How do I rotate a point clockwise about the origin?

A: To rotate a point clockwise about the origin, you need to follow these steps:

Reflect the point across the x-axis: When a point is reflected across the x-axis, its y-coordinate changes sign, while its x-coordinate remains the same.
Reflect the point across the y-axis: After reflecting the point across the x-axis, you need to reflect it again across the y-axis. This time, the x-coordinate changes sign, while the y-coordinate remains the same.

Q: How do I rotate a point 180° clockwise about the origin?

A: To rotate a point 180° clockwise about the origin, you need to follow these steps:

Reflect the point across the x-axis: When a point is reflected across the x-axis, its y-coordinate changes sign, while its x-coordinate remains the same.
Reflect the point across the y-axis: After reflecting the point across the x-axis, you need to reflect it again across the y-axis. This time, the x-coordinate changes sign, while the y-coordinate remains the same.

Q: How do I rotate a point 90° clockwise about the origin?

A: To rotate a point 90° clockwise about the origin, you need to follow these steps:

Reflect the point across the x-axis: When a point is reflected across the x-axis, its y-coordinate changes sign, while its x-coordinate remains the same.
Reflect the point across the y-axis: After reflecting the point across the x-axis, you need to reflect it again across the y-axis. This time, the x-coordinate changes sign, while the y-coordinate remains the same.

Q: How do I rotate a point 270° clockwise about the origin?

A: To rotate a point 270° clockwise about the origin, you need to follow these steps:

Reflect the point across the y-axis: When a point is reflected across the y-axis, its x-coordinate changes sign, while its y-coordinate remains the same.
Reflect the point across the x-axis: After reflecting the point across the y-axis, you need to reflect it again across the x-axis. This time, the y-coordinate changes sign, while the x-coordinate remains the same.