Rotation is a type of rigid body transformation which changes the orientation of the object. The most common type of rotation is a rotation around the origin. To look at how rotations are performed, let’s take *Point A,* which is located at the coordinates of *(a, b),** *as shown below:

Using the next image, the length of the yellow line is the value of *a*, and the length of the green line is the value of *b.*

To rotate the yellow and green lines, 90º around the origin gives us the following situation:

To find the coordinates of where the image, *point* *A’,* we see that the green line is now in the negative *x-direction,* and the yellow line is now in the *y-direction.* So, the coordinates of *A’* would be *(-b, a).*

This is the way to perform any 90º counterclockwise around the origin, you switch the values of the *x *and *y *coordinate values and make the new *x* value negative. If we wanted to perform a 180º counterclockwise rotation on around the origin on *point A*, would be the same as doing a 90º counterclockwise rotation *point A’. *So, the coordinates of this new point would be (-a, -b).

To perform a 270º counterclockwise rotation around the origin would move the point to coordinates *(b, -a).*

Use the table below to help you determine the coordinates of a point after a rotation was performed:

Rotation Amount | Direction | Coordinates |
---|---|---|

0º | Counterclockwise | (a, b) |

90º | Counterclockwise | (-b, a) |

180º | Counterclockwise | (-a, -b) |

270º | Counterclockwise | (b, -a) |

360º | Counterclockwise | (a, b) |

90º | Clockwise | (b, -a) |

180º | Clockwise | (-a, -b) |

270º | Clockwise | (-b, a) |

360º | Clockwise | (a, b) |