A Triangle Is Rotated 90 ∘ 90^{\circ} 9 0 ∘ About The Origin. Which Rule Describes The Transformation?A. { (x, Y) \rightarrow (-x, -y)$}$B. { (x, Y) \rightarrow (-y, X)$}$C. { (x, Y) \rightarrow (-y, -x)$} D . \[ D. \[ D . \[ (x, Y)

by ADMIN 233 views

Introduction

In geometry, transformations play a crucial role in understanding the properties and behavior of shapes. One of the fundamental transformations is rotation, where a shape is rotated by a certain angle around a fixed point, known as the origin. In this article, we will explore the transformation rule that describes a triangle rotated 90∘90^{\circ} about the origin.

Understanding Rotation

Rotation is a transformation that turns a shape around a fixed point, known as the origin. The origin is typically represented by the coordinates (0, 0). When a shape is rotated, its position and orientation change, but its size and shape remain the same.

The Rotation Matrix

To describe a rotation, we use a rotation matrix, which is a mathematical representation of the transformation. The rotation matrix for a 90∘90^{\circ} rotation about the origin is given by:

[0−110]\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

This matrix represents the transformation that takes a point (x, y) to its rotated position (-y, x).

The Transformation Rule

Now, let's analyze the transformation rule that describes a triangle rotated 90∘90^{\circ} about the origin. We can represent the transformation as a function that takes a point (x, y) to its rotated position.

The transformation rule can be represented as:

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

This rule states that the x-coordinate of the point is replaced by the negative of the y-coordinate, and the y-coordinate is replaced by the x-coordinate.

Analyzing the Options

Now, let's analyze the options given to determine which one describes the transformation rule.

Option A: $(x, y) \rightarrow (-x, -y)$

This option represents a reflection across the origin, not a rotation.

Option B: $(x, y) \rightarrow (-y, x)$

This option represents the transformation rule we derived earlier, which describes a 90∘90^{\circ} rotation about the origin.

Option C: $(x, y) \rightarrow (-y, -x)$

This option represents a reflection across the origin followed by a 90∘90^{\circ} rotation.

Option D: $(x, y) \rightarrow (y, x)$

This option represents a reflection across the y-axis, not a rotation.

Conclusion

In conclusion, the transformation rule that describes a triangle rotated 90∘90^{\circ} about the origin is:

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

This rule states that the x-coordinate of the point is replaced by the negative of the y-coordinate, and the y-coordinate is replaced by the x-coordinate.

Examples and Applications

Here are some examples and applications of the transformation rule:

  • Graphing: When graphing a function, we can use the transformation rule to rotate the graph by 90∘90^{\circ} about the origin.
  • Geometry: In geometry, we can use the transformation rule to describe the rotation of shapes, such as triangles and quadrilaterals.
  • Computer Graphics: In computer graphics, we can use the transformation rule to rotate objects in 2D and 3D space.

Exercises and Problems

Here are some exercises and problems to practice the transformation rule:

  • Exercise 1: Rotate the point (3, 4) by 90∘90^{\circ} about the origin using the transformation rule.
  • Exercise 2: Rotate the point (-2, 1) by 90∘90^{\circ} about the origin using the transformation rule.
  • Problem 1: Describe the transformation rule that describes a triangle rotated 180∘180^{\circ} about the origin.
  • Problem 2: Describe the transformation rule that describes a triangle rotated 270∘270^{\circ} about the origin.

Conclusion

In conclusion, the transformation rule that describes a triangle rotated 90∘90^{\circ} about the origin is:

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

This rule states that the x-coordinate of the point is replaced by the negative of the y-coordinate, and the y-coordinate is replaced by the x-coordinate. We can use this rule to describe the rotation of shapes in geometry, graphing, and computer graphics.

Introduction

In our previous article, we explored the transformation rule that describes a triangle rotated 90∘90^{\circ} about the origin. In this article, we will answer some frequently asked questions about the transformation rule and provide additional examples and explanations.

Q&A

Q: What is the transformation rule for a 90∘90^{\circ} rotation about the origin?

A: The transformation rule for a 90∘90^{\circ} rotation about the origin is:

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

This rule states that the x-coordinate of the point is replaced by the negative of the y-coordinate, and the y-coordinate is replaced by the x-coordinate.

Q: How do I apply the transformation rule to a point?

A: To apply the transformation rule to a point, simply replace the x-coordinate with the negative of the y-coordinate, and replace the y-coordinate with the x-coordinate.

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

A: A rotation is a transformation that turns a shape around a fixed point, known as the origin. A reflection is a transformation that flips a shape across a line or a point.

Q: Can I use the transformation rule to rotate a shape by an angle other than 90∘90^{\circ}?

A: Yes, you can use the transformation rule to rotate a shape by an angle other than 90∘90^{\circ}. However, you will need to use a different rotation matrix and apply the transformation rule accordingly.

Q: How do I graph a function after applying the transformation rule?

A: To graph a function after applying the transformation rule, simply substitute the transformed coordinates into the function.

Q: Can I use the transformation rule in computer graphics?

A: Yes, you can use the transformation rule in computer graphics to rotate objects in 2D and 3D space.

Q: What are some real-world applications of the transformation rule?

A: Some real-world applications of the transformation rule include:

  • Computer-Aided Design (CAD): The transformation rule is used in CAD software to rotate objects and shapes.
  • Computer Graphics: The transformation rule is used in computer graphics to rotate objects and shapes in 2D and 3D space.
  • Geometry: The transformation rule is used in geometry to describe the rotation of shapes and figures.

Examples and Applications

Here are some examples and applications of the transformation rule:

  • Graphing: When graphing a function, we can use the transformation rule to rotate the graph by 90∘90^{\circ} about the origin.
  • Geometry: In geometry, we can use the transformation rule to describe the rotation of shapes, such as triangles and quadrilaterals.
  • Computer Graphics: In computer graphics, we can use the transformation rule to rotate objects in 2D and 3D space.

Exercises and Problems

Here are some exercises and problems to practice the transformation rule:

  • Exercise 1: Rotate the point (3, 4) by 90∘90^{\circ} about the origin using the transformation rule.
  • Exercise 2: Rotate the point (-2, 1) by 90∘90^{\circ} about the origin using the transformation rule.
  • Problem 1: Describe the transformation rule that describes a triangle rotated 180∘180^{\circ} about the origin.
  • Problem 2: Describe the transformation rule that describes a triangle rotated 270∘270^{\circ} about the origin.

Conclusion

In conclusion, the transformation rule that describes a triangle rotated 90∘90^{\circ} about the origin is:

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

This rule states that the x-coordinate of the point is replaced by the negative of the y-coordinate, and the y-coordinate is replaced by the x-coordinate. We can use this rule to describe the rotation of shapes in geometry, graphing, and computer graphics.