Rotations

Unit 9: Transformations · ~15–30 min

What a rotation does

A rotation turns a figure about a fixed point, the center of rotation, through a given angle. Every point of the figure sweeps through the same angle at the same distance from the center — distances from the center and the shape of the figure are preserved, only the orientation in the plane changes. Rotations are described counterclockwise unless stated otherwise; a clockwise rotation of θ° is the same as a counterclockwise rotation of (360° − θ)°.

Coordinate rules about the origin

For rotations centered at the origin, three angles have simple coordinate rules:

• 90° counterclockwise: (x, y) → (−y, x)
• 180°: (x, y) → (−x, −y)
• 270° counterclockwise (= 90° clockwise): (x, y) → (y, −x)

Notice that 180° is its own reverse — rotating 180° twice returns a point to where it started. For a rotation about a point other than the origin, the same angle rules apply, but you first translate the center to the origin, rotate, then translate back.

Guided practice

• 1.Rotate point B(−4, 1) by 90° counterclockwise about the origin.
  Show answer

  (x, y) → (−y, x) gives B′(−1, −4).
• 2.Rotate point C(7, −2) by 180° about the origin.
  Show answer

  (x, y) → (−x, −y) gives C′(−7, 2).
• 3.Rotate point D(2, 6) by 270° counterclockwise about the origin.
  Show answer

  (x, y) → (y, −x) gives D′(6, −2).
• 4.Point E(−3, 4) maps to E′(3, −4). What rotation produced this?
  Show answer

  Both coordinates flipped sign with no swap, matching (x, y) → (−x, −y), so this is a 180° rotation.
• 5.Explain why a 180° rotation applied twice to any point returns it to its original position.
  Show answer

  Each 180° rotation sends (x, y) to (−x, −y). Applying it again sends (−x, −y) to (x, y) — the double sign flip cancels out, so the point lands back where it started.