By Allen Ma, Amber Kuang

In geometry, when you rotate an image, the sign of the degree of rotation tells you the direction in which the image is rotating. A positive degree measurement means you’re rotating counterclockwise, whereas a negative degree measurement means you’re rotating clockwise.

The following practice questions test your knowledge of rotations by asking you to rotate an octagon, and then to rotate coordinates of an image about the origin.

Practice questions

  1. The figure shows a regular octagon with l, m, and p as lines of symmetry.


    Find the image of the given point after the following rotation:


  2. Use your knowledge of rotations to answer this question: What are the coordinates of the image of (4, –6) after a clockwise rotation of 90 degrees about the origin?

Answers and explanations

  1. O

    When doing the composition of transformations, you perform the transformation closest to the point first. Therefore, you perform the transformations from right to left.

    Rotate Point A counterclockwise 405 degrees first. Rotating 405 degrees means you’re rotating more than a full revolution:


    This means you’re really just rotating the point 45 degrees counterclockwise:


    Take the new point and reflect it over line l:


    Take the new point and rotate it 135 degrees clockwise:


  2. (–6, –4)

    All the rules for rotations are written so that when you’re rotating counterclockwise, a full revolution is 360 degrees. Rotating 90 degrees clockwise is the same as rotating 270 degrees counterclockwise. Rotating 270 degrees counterclockwise about the origin is the same as reflecting over the line y = x and then reflecting over the x-axis. This means that the point (x, y) will become the point (y, –x). In this question, (4, –6) will become (–6, –4).