Moving a figure by turning it around a certain point is called a rotation. It is determined by the center of rotation and the angle and direction of rotation.
What Is a Rotation?
A rotation moves every point of a figure through the same angle and in the same direction around a point O (the center of rotation).
・The center O does not move (a fixed point).
・OA = OA'、OB = OB'、… (the distances from the center do not change)
・∠AOA' = ∠BOB' = ∠COC' = angle of rotation (every corresponding point rotates through the same angle)
Rotation Through 180° (Point Reflection)
A rotation through 180° is specifically called a point reflection.
・The center of rotation is the midpoint of the segment connecting corresponding points.
・The figure faces the opposite direction (it is not flipped over; its top, bottom, left, and right are exchanged).
Point A is rotated 90° counterclockwise about point O to create point A'. Answer the following questions.
(1) What is the relationship between the lengths of OA and OA'?
(2) What is the measure of ∠AOA'?
Solution:
(1) The distance from the center does not change in a rotation → OA = OA'
(2) The angle is the angle of rotation → 90°
Rotation Through 180° About the Origin in Coordinates
Point P' obtained by rotating point P(x, y) through 180° about the origin O is (−x, −y).
Example: P(3, −5) → P'(−3, 5).
Find the coordinates of point A' obtained by rotating point A(2, −3) through 180° about the origin O.
Solution:
A' = (−2, 3).
- Rotation: moving a figure through the same angle around one point O.
- The center of rotation O does not move, and distances from the center do not change.
- The angle of rotation is the same for every corresponding point.
- Rotation through 180° about the origin: (x, y) → (−x, −y).
Practice Problems
- When △ABC is rotated 60° about point O to create △A'B'C', find the measures of ∠AOA' and ∠BOB'.
- Find the coordinates of point P' obtained by rotating P(−4, 2) through 180° about the origin O.
- What is a rotation through 180° specifically called? Does the shape and size change under a 180° rotation?
Answers and Explanations
- Answer: Both are 60°
Explanation: The angle of rotation is the same for every corresponding point. - Answer: (4, −2)
Explanation: A 180° rotation gives (x, y) → (−x, −y). (−4, 2) → (4, −2). - Answer: Point reflection; the shape and size do not change
Explanation: A rotation preserves the shape and size. A 180° rotation is another name for point reflection.