Rotations About the Origin
A rotation turns every point of a figure around a fixed center through a given angle. This lesson covers the three standard counterclockwise rotations about the origin: 90°, 180°, and 270°.
Introduction
Imagine pinning a shape to the origin with a thumbtack and spinning it. The shape traces an arc, but every point stays the same distance from the center. By the end of this lesson you'll have three clean coordinate rules that handle the most common rotations instantly.
Past Knowledge
Transformation vocabulary (4.1.1). Reflections (4.1.4–4.1.5). Angle measure (1.3).
Today's Goal
Rotate points and figures 90°, 180°, and 270° counterclockwise about the origin.
Future Success
Non-origin rotations (4.1.7), rotational symmetry (4.2.1), and compositions (4.2.2).
Key Concepts
Standard Rotation Rules (Counterclockwise, Center = Origin)
| Angle (CCW) | Rule | Memory Aid |
|---|---|---|
| 90° | Swap, then negate the new | |
| 180° | Negate both | |
| 270° | Swap, then negate the new |
Clockwise vs. Counterclockwise
A 90° clockwise rotation gives the same result as a 270° counterclockwise rotation (and vice versa). In formula form: and .
180° — A Special Case
A 180° rotation is the same clockwise or counterclockwise. It's also equivalent to reflecting across both axes. The point ends up diagonally opposite the origin.
Key Properties of Rotations
- Rigid — distances and angles are preserved.
- Orientation preserved — unlike reflections, the “handedness” stays the same.
- The center of rotation is the only fixed point (it maps to itself).
- Each point and its image are the same distance from the center.
Worked Examples
90° CCW Rotation
Rotate by 90° counterclockwise about the origin.
Rule:
180° Rotation of a Triangle
Rotate with , , by 180° about the origin.
Rule:
90° Clockwise (= 270° CCW)
Rotate with , , by 90° clockwise about the origin.
90° CW = 270° CCW, so use the 270° rule:
Common Pitfalls
Mixing Up 90° and 270° Rules
90° CCW: . 270° CCW: . The negative sign goes on different coordinates — practice until the pattern is automatic.
Forgetting That “Clockwise” and “Counterclockwise” Give Different Images
90° CW ≠ 90° CCW! Always check whether the problem says clockwise or counterclockwise. Clockwise 90° = counterclockwise 270°.
Sign Errors with Negative Coordinates
If and you rotate 90° CCW: . Be very careful negating a coordinate that's already negative.
Real-Life Applications
Clocks & Gears
Clock hands rotate about the center of the clock face. The minute hand completes a 360° rotation every hour, while each hour mark is positioned at 30° increments — all governed by the same rotation math you learned today.
Robotics & Drone Navigation
When a drone or robot arm pivots around a joint, engineers calculate the new position of each component using rotation matrices — a direct extension of the coordinate rules in this lesson.
Practice Quiz
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