Triangle QRS Has Vertices \[$ Q(-2, 2), R(-3, -4), \$\] And \[$ S(1, -2) \$\].Write The Coordinate Notation For Each Rotation Given. Then Write The Coordinates Of \[$\triangle Q^{\prime} R^{\prime} S^{\prime}\$\] After Each

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Introduction

In geometry, rotation is a fundamental concept that involves turning a figure around a fixed point or axis. In this article, we will explore the rotation of a triangle QRS with vertices Q(-2, 2), R(-3, -4), and S(1, -2) around the origin (0, 0). We will write the coordinate notation for each rotation and determine the coordinates of the rotated triangle Q'R'S'.

Rotation of 90 Degrees Counterclockwise

A rotation of 90 degrees counterclockwise around the origin can be represented by the following coordinate notation:

  • Q' = (2, 2)
  • R' = (4, -3)
  • S' = (-2, -2)

To obtain these coordinates, we can use the following rotation formulas:

  • x' = -y
  • y' = x

Applying these formulas to the original coordinates of Q, R, and S, we get:

  • Q' = (-2, 2) -> (2, 2)
  • R' = (-3, -4) -> (4, -3)
  • S' = (1, -2) -> (-2, -2)

Rotation of 180 Degrees Counterclockwise

A rotation of 180 degrees counterclockwise around the origin can be represented by the following coordinate notation:

  • Q' = (-2, -2)
  • R' = (3, 4)
  • S' = (-1, 2)

To obtain these coordinates, we can use the following rotation formulas:

  • x' = -x
  • y' = -y

Applying these formulas to the original coordinates of Q, R, and S, we get:

  • Q' = (-2, 2) -> (-2, -2)
  • R' = (-3, -4) -> (3, 4)
  • S' = (1, -2) -> (-1, 2)

Rotation of 270 Degrees Counterclockwise

A rotation of 270 degrees counterclockwise around the origin can be represented by the following coordinate notation:

  • Q' = (-2, -2)
  • R' = (-4, 3)
  • S' = (2, 2)

To obtain these coordinates, we can use the following rotation formulas:

  • x' = y
  • y' = -x

Applying these formulas to the original coordinates of Q, R, and S, we get:

  • Q' = (-2, 2) -> (-2, -2)
  • R' = (-3, -4) -> (-4, 3)
  • S' = (1, -2) -> (2, 2)

Rotation of 90 Degrees Clockwise

A rotation of 90 degrees clockwise around the origin can be represented by the following coordinate notation:

  • Q' = (-2, -2)
  • R' = (-4, 3)
  • S' = (2, 2)

To obtain these coordinates, we can use the following rotation formulas:

  • x' = y
  • y' = -x

Applying these formulas to the original coordinates of Q, R, and S, we get:

  • Q' = (-2, 2) -> (-2, -2)
  • R' = (-3, -4) -> (-4, 3)
  • S' = (1, -2) -> (2, 2)

Rotation of 180 Degrees Clockwise

A rotation of 180 degrees clockwise around the origin can be represented by the following coordinate notation:

  • Q' = (-2, -2)
  • R' = (3, 4)
  • S' = (-1, 2)

To obtain these coordinates, we can use the following rotation formulas:

  • x' = -x
  • y' = -y

Applying these formulas to the original coordinates of Q, R, and S, we get:

  • Q' = (-2, 2) -> (-2, -2)
  • R' = (-3, -4) -> (3, 4)
  • S' = (1, -2) -> (-1, 2)

Rotation of 270 Degrees Clockwise

A rotation of 270 degrees clockwise around the origin can be represented by the following coordinate notation:

  • Q' = (2, 2)
  • R' = (4, -3)
  • S' = (-2, -2)

To obtain these coordinates, we can use the following rotation formulas:

  • x' = -y
  • y' = x

Applying these formulas to the original coordinates of Q, R, and S, we get:

  • Q' = (-2, 2) -> (2, 2)
  • R' = (-3, -4) -> (4, -3)
  • S' = (1, -2) -> (-2, -2)

Conclusion

In this article, we have explored the rotation of a triangle QRS with vertices Q(-2, 2), R(-3, -4), and S(1, -2) around the origin (0, 0). We have written the coordinate notation for each rotation and determined the coordinates of the rotated triangle Q'R'S'. The rotation formulas used in this article are:

  • x' = -y
  • y' = x
  • x' = -x
  • y' = -y
  • x' = y
  • y' = -x

Q: What is the purpose of rotating a triangle?

A: Rotating a triangle is a fundamental concept in geometry that helps us understand how shapes change when they are turned around a fixed point or axis. It is used in various fields such as art, architecture, engineering, and computer graphics.

Q: What are the different types of rotations?

A: There are two main types of rotations: clockwise and counterclockwise. A clockwise rotation is a rotation that turns a shape in a clockwise direction, while a counterclockwise rotation is a rotation that turns a shape in a counterclockwise direction.

Q: How do you rotate a point in a coordinate plane?

A: To rotate a point in a coordinate plane, you can use the following rotation formulas:

  • x' = -y
  • y' = x
  • x' = -x
  • y' = -y
  • x' = y
  • y' = -x

These formulas can be used to rotate any point or figure around the origin in a coordinate plane.

Q: What is the effect of rotating a triangle on its coordinates?

A: When a triangle is rotated, its coordinates change. The new coordinates of the triangle can be found by applying the rotation formulas to the original coordinates of the triangle.

Q: Can you give an example of rotating a triangle?

A: Yes, let's consider the triangle QRS with vertices Q(-2, 2), R(-3, -4), and S(1, -2). If we rotate this triangle 90 degrees counterclockwise around the origin, the new coordinates of the triangle will be:

  • Q' = (2, 2)
  • R' = (4, -3)
  • S' = (-2, -2)

Q: How do you determine the coordinates of a rotated triangle?

A: To determine the coordinates of a rotated triangle, you can use the rotation formulas and apply them to the original coordinates of the triangle.

Q: What are some real-world applications of rotating triangles?

A: Rotating triangles has many real-world applications in various fields such as:

  • Art: Rotating triangles is used in art to create symmetrical and asymmetrical shapes.
  • Architecture: Rotating triangles is used in architecture to design buildings and structures.
  • Engineering: Rotating triangles is used in engineering to design machines and mechanisms.
  • Computer Graphics: Rotating triangles is used in computer graphics to create 3D models and animations.

Q: Can you summarize the key points of rotating triangles?

A: Yes, the key points of rotating triangles are:

  • Rotating a triangle is a fundamental concept in geometry.
  • There are two main types of rotations: clockwise and counterclockwise.
  • The rotation formulas are used to rotate a point or figure around the origin in a coordinate plane.
  • The new coordinates of a rotated triangle can be found by applying the rotation formulas to the original coordinates of the triangle.
  • Rotating triangles has many real-world applications in various fields.