A Segment Has Endpoints $X(-10,0$\] And $Y(-2,6$\]. Consider Its Image After A $180^{\circ}$ (counterclockwise) Rotation About The Origin. Select The Coordinates Of $Y^{\prime}$.A. $Y^{\prime}(2,-6$\]B.

Feb 25, 2025 by ADMIN 203 views

A Segment Has Endpoints X(-10,0) and Y(-2,6): Finding the Image After a 180° Counterclockwise Rotation About the Origin

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 180° counterclockwise about the origin, its coordinates are negated. In this article, we will explore how to find the image of a segment after a 180° counterclockwise rotation about the origin.

Understanding the Problem

We are given a segment with endpoints X(-10,0) and Y(-2,6). The problem asks us to find the coordinates of Y' after a 180° counterclockwise rotation about the origin.

Rotating a Point 180° Counterclockwise About the Origin

To rotate a point 180° counterclockwise about the origin, we need to negate its coordinates. This means that the x-coordinate becomes its opposite, and the y-coordinate becomes its opposite.

Finding the Image of Y

To find the image of Y, we need to negate its coordinates. The coordinates of Y are (-2,6). Negating these coordinates, we get (2,-6).

Conclusion

In conclusion, after a 180° counterclockwise rotation about the origin, the coordinates of Y' are (2,-6).

Step-by-Step Solution

Here's a step-by-step solution to the problem:

Identify the coordinates of Y: The coordinates of Y are (-2,6).
Negate the coordinates of Y: To rotate Y 180° counterclockwise about the origin, we need to negate its coordinates. This means that the x-coordinate becomes its opposite, and the y-coordinate becomes its opposite.
Find the image of Y: The image of Y is obtained by negating its coordinates. Therefore, the coordinates of Y' are (2,-6).

Example

Let's consider an example to illustrate the concept. Suppose we have a point P(3,4) and we want to rotate it 180° counterclockwise about the origin. To do this, we need to negate its coordinates. The coordinates of P are (3,4). Negating these coordinates, we get (-3,-4). Therefore, the image of P after a 180° counterclockwise rotation about the origin is (-3,-4).

Applications

Rotations have many applications in mathematics and real-life situations. For example, in computer graphics, rotations are used to create 3D models and animations. In engineering, rotations are used to design and analyze mechanical systems.

Summary

In this article, we explored how to find the image of a segment after a 180° counterclockwise rotation about the origin. We learned that to rotate a point 180° counterclockwise about the origin, we need to negate its coordinates. We applied this concept to find the image of Y and provided a step-by-step solution to the problem.

Frequently Asked Questions

What is a 180° counterclockwise rotation about the origin? A 180° counterclockwise rotation about the origin is a transformation that turns a figure around the origin by 180° in a counterclockwise direction.
How do I find the image of a point after a 180° counterclockwise rotation about the origin? To find the image of a point after a 180° counterclockwise rotation about the origin, you need to negate its coordinates.
What are the coordinates of Y' after a 180° counterclockwise rotation about the origin? The coordinates of Y' after a 180° counterclockwise rotation about the origin are (2,-6).

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Mathematics for Computer Graphics" by Eric Haines
[3] "Engineering Mathematics" by K.A. Stroud

Keywords

180° counterclockwise rotation about the origin
Negating coordinates
Image of a point
Geometry
Computer graphics
Engineering mathematics

Introduction

In our previous article, we explored how to find the image of a segment after a 180° counterclockwise rotation about the origin. We learned that to rotate a point 180° counterclockwise about the origin, we need to negate its coordinates. In this article, we will answer some frequently asked questions on 180° counterclockwise rotation about the origin.

Q&A

Q1: What is a 180° counterclockwise rotation about the origin?

A1: A 180° counterclockwise rotation about the origin is a transformation that turns a figure around the origin by 180° in a counterclockwise direction.

Q2: How do I find the image of a point after a 180° counterclockwise rotation about the origin?

A2: To find the image of a point after a 180° counterclockwise rotation about the origin, you need to negate its coordinates.

Q3: What are the coordinates of Y' after a 180° counterclockwise rotation about the origin?

A3: The coordinates of Y' after a 180° counterclockwise rotation about the origin are (2,-6).

Q4: Can I rotate a point 180° clockwise about the origin?

A4: No, you cannot rotate a point 180° clockwise about the origin. The rotation is always counterclockwise.

Q5: How do I rotate a point 180° counterclockwise about a point other than the origin?

A5: To rotate a point 180° counterclockwise about a point other than the origin, you need to translate the point to the origin, rotate it 180° counterclockwise, and then translate it back.

Q6: What is the effect of a 180° counterclockwise rotation about the origin on the x and y coordinates?

A6: A 180° counterclockwise rotation about the origin negates the x and y coordinates.

Q7: Can I rotate a point 180° counterclockwise about the origin using a matrix?

A7: Yes, you can rotate a point 180° counterclockwise about the origin using a matrix. The matrix is:

[ -1  0 ]
[  0 -1 ]

Q8: How do I find the image of a line after a 180° counterclockwise rotation about the origin?

A8: To find the image of a line after a 180° counterclockwise rotation about the origin, you need to rotate the line 180° counterclockwise.

Q9: Can I rotate a point 180° counterclockwise about the origin using a rotation formula?

A9: Yes, you can rotate a point 180° counterclockwise about the origin using a rotation formula. The formula is:

x' = -x
y' = -y

Q10: What is the effect of a 180° counterclockwise rotation about the origin on the distance from the origin?

A10: A 180° counterclockwise rotation about the origin does not change the distance from the origin.