A Rectangle Is Transformed According To The Rule \[$ R_{0,90^{\circ}} \$\]. The Image Of The Rectangle Has Vertices Located At \[$ R^{\prime}(-4,4) \$\], \[$ S^{\prime}(-4,1) \$\], \[$ P^{\prime}(-3,1) \$\], And \[$

Feb 28, 2025 by ADMIN 216 views

A Rectangle Transformed by a 90-Degree Rotation

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Introduction

In geometry, a transformation is a way of changing the position or size of a shape. One common type of transformation is a rotation, which involves turning a shape around a fixed point. In this article, we will explore the transformation of a rectangle by a 90-degree rotation.

The Rule of Rotation

The rule of rotation ${$ R_{0,90^{\circ}} $}$ involves rotating a shape by 90 degrees around the origin (0,0). This means that each point on the shape will be moved to a new location that is 90 degrees counterclockwise from its original position.

The Image of the Rectangle

The image of the rectangle has vertices located at ${$ R^{\prime}(-4,4) $}$, ${$ S^{\prime}(-4,1) $}$, ${$ P^{\prime}(-3,1) $}$, and ${$ Q^{\prime}(-3,4) $}$. To find the image of the rectangle, we need to apply the rule of rotation to each of its vertices.

Applying the Rule of Rotation

To apply the rule of rotation, we need to multiply each vertex by the rotation matrix ${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} $}$. This matrix represents a 90-degree counterclockwise rotation around the origin.

Vertex R

The vertex R has coordinates (0,0). To find the image of R, we multiply its coordinates by the rotation matrix:

${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 \ 0 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix} $}$

The image of R is (0,0).

Vertex S

The vertex S has coordinates (0,1). To find the image of S, we multiply its coordinates by the rotation matrix:

${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 \ 1 \end{bmatrix} = \begin{bmatrix} -1 \ 0 \end{bmatrix} $}$

The image of S is (-1,0).

Vertex P

The vertex P has coordinates (1,0). To find the image of P, we multiply its coordinates by the rotation matrix:

${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \ 0 \end{bmatrix} = \begin{bmatrix} 0 \ 1 \end{bmatrix} $}$

The image of P is (0,1).

Vertex Q

The vertex Q has coordinates (1,1). To find the image of Q, we multiply its coordinates by the rotation matrix:

${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \ 1 \end{bmatrix} = \begin{bmatrix} -1 \ 1 \end{bmatrix} $}$

The image of Q is (-1,1).

Conclusion

In this article, we explored the transformation of a rectangle by a 90-degree rotation. We applied the rule of rotation to each vertex of the rectangle and found its image. The image of the rectangle has vertices located at ${$ R^{\prime}(-4,4) $}$, ${$ S^{\prime}(-4,1) $}$, ${$ P^{\prime}(-3,1) $}$, and ${$ Q^{\prime}(-3,4) $}$. This transformation can be used to solve problems in geometry and trigonometry.

Applications of Rotation

Rotation is an important concept in geometry and trigonometry. It has many applications in real-life situations, such as:

Design and Architecture: Rotation is used in the design of buildings, bridges, and other structures to create symmetrical and aesthetically pleasing shapes.
Engineering: Rotation is used in the design of machines and mechanisms to create efficient and effective systems.
Computer Graphics: Rotation is used in computer graphics to create 3D models and animations.
Navigation: Rotation is used in navigation systems to determine the direction and position of objects.

Exercises

Problem 1: A rectangle has vertices located at (0,0), (1,0), (1,1), and (0,1). Apply a 90-degree rotation to the rectangle and find its image.
Problem 2: A triangle has vertices located at (0,0), (1,0), and (0,1). Apply a 90-degree rotation to the triangle and find its image.
Problem 3: A square has vertices located at (0,0), (1,0), (1,1), and (0,1). Apply a 90-degree rotation to the square and find its image.

Solutions

Solution 1: To apply a 90-degree rotation to the rectangle, we multiply each vertex by the rotation matrix ${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} $}$. The image of the rectangle has vertices located at (0,0), (-1,0), (0,-1), and (1,-1).
Solution 2: To apply a 90-degree rotation to the triangle, we multiply each vertex by the rotation matrix ${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} $}$. The image of the triangle has vertices located at (0,0), (-1,0), and (0,-1).
Solution 3: To apply a 90-degree rotation to the square, we multiply each vertex by the rotation matrix ${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} $}$. The image of the square has vertices located at (0,0), (-1,0), (0,-1), and (1,-1).

Conclusion

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Introduction

In our previous article, we explored the transformation of a rectangle by a 90-degree rotation. We applied the rule of rotation to each vertex of the rectangle and found its image. In this article, we will answer some frequently asked questions about the transformation of a rectangle by a 90-degree rotation.

Q&A

Q1: What is a 90-degree rotation?

A1: A 90-degree rotation is a transformation that involves turning a shape by 90 degrees around the origin (0,0). This means that each point on the shape will be moved to a new location that is 90 degrees counterclockwise from its original position.

Q2: How do I apply a 90-degree rotation to a rectangle?

A2: To apply a 90-degree rotation to a rectangle, you need to multiply each vertex of the rectangle by the rotation matrix ${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} $}$. This matrix represents a 90-degree counterclockwise rotation around the origin.

Q3: What is the image of a rectangle after a 90-degree rotation?

A3: The image of a rectangle after a 90-degree rotation is a new rectangle with vertices located at the rotated positions of the original rectangle's vertices.

Q4: Can I apply a 90-degree rotation to any shape?

A4: Yes, you can apply a 90-degree rotation to any shape. However, the shape must be defined by a set of vertices, and the rotation matrix must be applied to each vertex.

Q5: How do I find the image of a shape after a 90-degree rotation?

A5: To find the image of a shape after a 90-degree rotation, you need to multiply each vertex of the shape by the rotation matrix ${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} $}$. This will give you the new coordinates of the shape's vertices.

Q6: Can I use a 90-degree rotation to solve problems in geometry and trigonometry?

A6: Yes, you can use a 90-degree rotation to solve problems in geometry and trigonometry. Rotation is an important concept in these subjects, and it has many applications in real-life situations.

Q7: What are some real-life applications of a 90-degree rotation?

A7: Some real-life applications of a 90-degree rotation include:

Design and Architecture: Rotation is used in the design of buildings, bridges, and other structures to create symmetrical and aesthetically pleasing shapes.
Engineering: Rotation is used in the design of machines and mechanisms to create efficient and effective systems.
Computer Graphics: Rotation is used in computer graphics to create 3D models and animations.
Navigation: Rotation is used in navigation systems to determine the direction and position of objects.

Conclusion

In this article, we answered some frequently asked questions about the transformation of a rectangle by a 90-degree rotation. We explained what a 90-degree rotation is, how to apply it to a rectangle, and what the image of a rectangle is after a 90-degree rotation. We also discussed some real-life applications of a 90-degree rotation and how it can be used to solve problems in geometry and trigonometry.

Exercises

Problem 1: A rectangle has vertices located at (0,0), (1,0), (1,1), and (0,1). Apply a 90-degree rotation to the rectangle and find its image.
Problem 2: A triangle has vertices located at (0,0), (1,0), and (0,1). Apply a 90-degree rotation to the triangle and find its image.
Problem 3: A square has vertices located at (0,0), (1,0), (1,1), and (0,1). Apply a 90-degree rotation to the square and find its image.

Solutions

Solution 1: To apply a 90-degree rotation to the rectangle, you need to multiply each vertex by the rotation matrix ${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} $}$. The image of the rectangle has vertices located at (0,0), (-1,0), (0,-1), and (1,-1).
Solution 2: To apply a 90-degree rotation to the triangle, you need to multiply each vertex by the rotation matrix ${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} $}$. The image of the triangle has vertices located at (0,0), (-1,0), and (0,-1).
Solution 3: To apply a 90-degree rotation to the square, you need to multiply each vertex by the rotation matrix ${$ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} $}$. The image of the square has vertices located at (0,0), (-1,0), (0,-1), and (1,-1).