Rotations

Standard

G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Goals for this section:

To understand that rotations will generally go in the counterclockwise direction unless specified and that in this class, we'll generally go 90, 180, or 270 degrees.

This notation means: the rotation going x° counterclockwise of UVW about the point Q is U'V'W'

  • r stands for rotation
  • x° how much you'll rotate
  • Q is where you're rotating around

Rotating 90°, 180°, and 270° counterclockwise about the origin

r(90°,O)(x,y)=(-y,x)

This is how to apply a 90° rotation about the origin which is O.

  • Notice how the lines make a 90° angle about the origin.
  • (1,2) is what we're rotating.
  • Take the y-value, 2, change its sign, and put it in the first spot like this (-2,_).
  • Now take the x-value, 1, and put it in the second spot like this (-2, 1).

r(180°,O)(x,y)=(-x,-y)

This is how to apply a 180° rotation about the origin which is O.

  • Notice how the lines make a 180° angle about the origin.
  • (1,2) is what we're rotating.
  • Take the x-value, 1, and change the sign. Our coordinate will look like (-1,_).
  • Take the y-value, 2, and change the sign. Our coordinate will be (-1, -2).

r(270°,O)(x,y)=(y,-x)

This is how to apply a 270° rotation about the origin which is O.

  • Notice how the lines make a 270° angle about the origin.
  • (1,2) is what we're rotating.
  • Take the y-value, 2, and put it in the first spot like this (2,_).
  • Take the x-value, 1, change the sign and put it in the second spot like this (2, -1).

Rotating Clockwise

Rotating clockwise is the same as counterclockwise

So notice how if you're asked to rotate clockwise 270°, it's the same as rotating counterclockwise 90°.

Similarly,

  • Rotating clockwise 180° is the same as rotating counterclockwise 180°.
  • Rotating clockwise 90° is the same as rotating counterclockwise 270°

Keywords: rotations, 90, 180, 270, counterclockwise, clockwise