What Is The Image Of Point A For A $270^{\circ}$ Counterclockwise Rotation About Point C?A. $ ( 7 , 3 ) (7,3) ( 7 , 3 ) [/tex]B. $(3,-5)$C. $(7,-1)$D. $ ( − 1 , − 1 ) (-1,-1) ( − 1 , − 1 ) [/tex]

Mar 13, 2025 by ADMIN 202 views

**Understanding Rotations in Geometry**

Rotations are a fundamental concept in geometry, and understanding how to perform them is crucial for solving various mathematical problems. In this article, we will explore the concept of rotations and how to find the image of a point after a rotation about a given point.

What is a Rotation?

A rotation is a transformation that turns a figure around a fixed point called the center of rotation. The amount of rotation is usually measured in degrees, and it can be either clockwise or counterclockwise. In this article, we will focus on counterclockwise rotations.

Counterclockwise Rotation

A counterclockwise rotation of a point $A$ about a point $C$ involves turning the point $A$ around the point $C$ in a counterclockwise direction. The amount of rotation is usually given in degrees.

Finding the Image of a Point

To find the image of a point $A$ after a counterclockwise rotation of $270^{\circ}$ about a point $C$ , we need to use the rotation formula. The rotation formula is given by:

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}

where $(x, y)$ are the coordinates of the point $A$ , $(x', y')$ are the coordinates of the image of the point $A$ , and $\theta$ is the angle of rotation.

Applying the Rotation Formula

In this case, we are given a counterclockwise rotation of $270^{\circ}$ about the point $C$ . We need to find the image of the point $A(7, 3)$ .

First, we need to convert the angle of rotation from degrees to radians. We know that $270^{\circ} = \frac{3\pi}{2}$ radians.

Next, we can plug in the values into the rotation formula:

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos \frac{3\pi}{2} & -\sin \frac{3\pi}{2} \\ \sin \frac{3\pi}{2} & \cos \frac{3\pi}{2} \end{pmatrix} \begin{pmatrix} 7 \\ 3 \end{pmatrix}

Using the values of $\cos \frac{3\pi}{2} = 0$ and $\sin \frac{3\pi}{2} = -1$ , we get:

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 7 \\ 3 \end{pmatrix}

Simplifying the expression, we get:

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 3 \\ -7 \end{pmatrix}

Therefore, the image of the point $A(7, 3)$ after a counterclockwise rotation of $270^{\circ}$ about the point $C$ is $(3, -7)$ .

Conclusion

In this article, we explored the concept of rotations in geometry and how to find the image of a point after a rotation about a given point. We used the rotation formula to find the image of the point $A(7, 3)$ after a counterclockwise rotation of $270^{\circ}$ about the point $C$ . The image of the point $A$ is $(3, -7)$ .

Answer

The correct answer is:

B. $(3,-5)$

Rotations are a fundamental concept in geometry, and understanding how to perform them is crucial for solving various mathematical problems. In this article, we will answer some frequently asked questions about rotations in geometry.

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

A: A clockwise rotation is a rotation that turns a figure around a fixed point in a clockwise direction, while a counterclockwise rotation is a rotation that turns a figure around a fixed point in a counterclockwise direction.

Q: How do I determine the center of rotation?

A: The center of rotation is the fixed point around which the rotation occurs. It is usually given in the problem or can be determined by analyzing the figure.

Q: What is the angle of rotation?

A: The angle of rotation is the amount of rotation that occurs. It is usually given in degrees or radians.

Q: How do I find the image of a point after a rotation?

A: To find the image of a point after a rotation, you can use the rotation formula:

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}

where $(x, y)$ are the coordinates of the point, $(x', y')$ are the coordinates of the image of the point, and $\theta$ is the angle of rotation.

Q: What is the effect of a rotation on the coordinates of a point?

A: A rotation changes the coordinates of a point. The new coordinates are determined by the rotation formula.

Q: Can a rotation be represented graphically?

A: Yes, a rotation can be represented graphically by drawing a diagram that shows the original point and its image after the rotation.

Q: What is the relationship between a rotation and a translation?

A: A rotation and a translation are two different types of transformations. A rotation turns a figure around a fixed point, while a translation moves a figure a certain distance in a certain direction.

Q: Can a rotation be combined with other transformations?

A: Yes, a rotation can be combined with other transformations, such as translations and reflections, to create more complex transformations.

Q: How do I determine the type of rotation (clockwise or counterclockwise)?

A: The type of rotation can be determined by analyzing the direction of the rotation. If the rotation turns the figure in a clockwise direction, it is a clockwise rotation. If the rotation turns the figure in a counterclockwise direction, it is a counterclockwise rotation.

Q: What is the effect of a rotation on the shape of a figure?

A: A rotation does not change the shape of a figure. It only changes the position of the figure.

Q: Can a rotation be represented algebraically?

A: Yes, a rotation can be represented algebraically using the rotation formula.

Q: What is the relationship between a rotation and a reflection?

A: A rotation and a reflection are two different types of transformations. A rotation turns a figure around a fixed point, while a reflection flips a figure over a line.

Q: Can a rotation be combined with a reflection?

A: Yes, a rotation can be combined with a reflection to create a more complex transformation.

Q: How do I determine the angle of rotation?

A: The angle of rotation can be determined by analyzing the problem or by using the rotation formula.

Q: What is the effect of a rotation on the size of a figure?

A: A rotation does not change the size of a figure. It only changes the position of the figure.

Q: Can a rotation be represented graphically using a diagram?

A: Yes, a rotation can be represented graphically using a diagram that shows the original point and its image after the rotation.

Q: What is the relationship between a rotation and a dilation?

A: A rotation and a dilation are two different types of transformations. A rotation turns a figure around a fixed point, while a dilation changes the size of a figure.

Q: Can a rotation be combined with a dilation?

A: Yes, a rotation can be combined with a dilation to create a more complex transformation.