Given a point on a geometric figure (x, y) , what is the new point when the figure is rotated clockwise about the origin 270 degrees?

Answer 1

To find the new point after rotating a given point (x, y) clockwise about the origin by 270 degrees, we can use the rotation matrix.

The rotation matrix for rotating a point (x, y) clockwise by θ degrees is given by:

| cosθ -sinθ |
| sinθ cosθ |

In this case, θ = 270 degrees.

So, substituting θ = 270 degrees, the rotation matrix becomes:

| cos270 -sin270 |
| sin270 cos270 |

To find the new point, we multiply the rotation matrix by the coordinates of the given point:

| cos270 -sin270 | | x |
| sin270 cos270 | * | y |

Calculating the values:

| 0 -1 | | x |
| 1 0 | * | y |

Multiplying the matrices, we get:

New x-coordinate: 0 * x + (-1) * y = -y
New y-coordinate: 1 * x + 0 * y = x

Therefore, the new point after rotating clockwise about the origin by 270 degrees is (-y, x).

Answer 2

Given a point (1, 2) on a geometric figure, what is the new point when the figure is rotated counterclockwise about the origin 90 degrees?

Answer 3

To find the new point after rotating a given point (1, 2) counterclockwise about the origin by 90 degrees, we can use the rotation matrix.

The rotation matrix for rotating a point (x, y) counterclockwise by θ degrees is given by:

| cosθ -sinθ |
| sinθ cosθ |

In this case, θ = 90 degrees.

So, substituting θ = 90 degrees, the rotation matrix becomes:

| cos90 -sin90 |
| sin90 cos90 |

To find the new point, we multiply the rotation matrix by the coordinates of the given point:

| cos90 -sin90 | | 1 |
| sin90 cos90 | * | 2 |

Calculating the values:

| 0 -1 | | 1 |
| 1 0 | * | 2 |

Multiplying the matrices, we get:

New x-coordinate: 0 * 1 + (-1) * 2 = -2
New y-coordinate: 1 * 1 + 0 * 2 = 1

Therefore, the new point after rotating counterclockwise about the origin by 90 degrees is (-2, 1).