16.2: Rotations

Links and useful resources

• gr10's Geometry >>>START HERE<<<
• AoPS Online Textbook
• AoPS Alcumus
• Big Ideas Geometry textbook
• GeoGebra Online Geometry Constuction Tool
• Two-column math templates
• Proofs unit slides from mathgiraffe.com
• Proofs unit printables for two-column proofs

Concept summary and connections

• A rotation of $\theta$ about a point $O$ makes a congruent image of the original subject, such that any segment drawn from $O$ to a point on the original will be at angle $\theta$ to a segment from $O$ to the corresponding point on the image.
• A rotation of $360\mathrm{°}$ always makes an image that coincides with the original.
• Rotations can be more than $360\mathrm{°}$! Don't forget that when you do your problems!

Lesson and worked examples

A rotation is a transformation that:

• produces an image that is congruent to the original thing being rotated
• can change both the orientation and position of the diagram (you can rotate about a point outside of the diagram, and that will make it move)
• "spins" the diagram by a specific angle.

Rotations always have a center point and an angle, and in 2D can be either clockwise or counterclockwise. By convention, a positive angle rotation is counterclockwise. This is just an arbitrary choice, but it was made long ago and people stick with it to make it easier to communicate.

Guided practice

• After I draw a figure, I rotate it ${48}^{\circ }$ about some point $P$. If the image of this rotation coincides exactly with the original figure, is it true that the image of a rotation of the figure ${72}^{\circ }$ about $P$ must also coincide with the figure?
• Through how many different positive angles $\theta$ less than ${360}^{\circ }$ is it possible to rotate a regular dodecagon clockwise about its center such that its image coincides with the original dodecagon?

Homework

• #hw (geom-gr10) 16.2 Rotations (p. 411) (gr10) 2025-03-28