Rotations
A rotation turns a figure around a fixed point, like a wheel turning on its hub. Every point stays the same distance from the center.
Know these first
- Reflections 8.G.1
- Master List skill 6.NS.6
This unlocks
What you'll be able to do
- Rotate points and figures 90°, 180° and 270° about the origin.
- Tell clockwise from counterclockwise and switch between them.
- Find the rotation from a point and its image.
- Rotate about a point other than the origin.
The big idea
A rotation needs three things: a center, an angle and a direction. Counterclockwise is the standard direction unless you're told otherwise.
About the origin, the rules are short: 90° counterclockwise (x, y) → (−y, x); 180° (x, y) → (−x, −y); 270° counterclockwise (x, y) → (y, −x). A 90° clockwise turn is the same as 270° counterclockwise.
To rotate about another point, slide everything so that point sits on the origin, rotate, then slide back. Rotations keep lengths, angles and orientation.
Words you need
- Center of rotation
- The fixed point everything turns around.
- Counterclockwise
- The opposite way to a clock's hands - the standard direction.
- Quarter-turn
- A 90° rotation. Four quarter-turns bring a figure back to the start.
Rotation rules about the origin
| Turn | Rule | Example |
|---|---|---|
| 90° counterclockwise | (x, y) → (−y, x) | (3, 1) → (−1, 3) |
| 180° | (x, y) → (−x, −y) | (3, 1) → (−3, −1) |
| 270° counterclockwise = 90° clockwise | (x, y) → (y, −x) | (3, 1) → (1, −3) |
Worked examples
Example 1
Rotate the point (3, 1) 90° counterclockwise about the origin.
- Use (x, y) → (−y, x).
- −y = −1 and x = 3, so the image is (−1, 3).
Answer: (−1, 3)
Example 2
Rotate the point (2, −4) 180° about the origin.
- A half-turn changes the sign of both coordinates: (x, y) → (−x, −y).
- (2, −4) → (−2, 4).
Answer: (−2, 4)
Example 3
Rotate triangle ABC with A(1, 1), B(4, 1) and C(1, 3) 90° clockwise about the origin.
- 90° clockwise is the same as 270° counterclockwise: (x, y) → (y, −x).
- A(1, 1) → (1, −1), B(4, 1) → (1, −4), C(1, 3) → (3, −1).
Answer: (1, −1), (1, −4), (3, −1)
Common mistakes
Mixing up the 90° rules. Counterclockwise is (−y, x). Check with an easy point: (1, 0) should turn to (0, 1), straight up.
Forgetting which way is clockwise. Clockwise follows a clock's hands: from 12 to 3. 90° clockwise = 270° counterclockwise.
Using the origin when the center is elsewhere. Shift so the center is at the origin, rotate, then shift back.
Practice: 20 questions
12 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Rotate the point (4, 2) 180° about the origin.
- Rotate the point (1, 5) 90° counterclockwise about the origin.
- A windmill blade's tip is at (0, 6). It turns 90° counterclockwise about the hub at the origin. Where is the tip now? Word problem
- The tip of a clock hand is at (0, 4). It turns 90° clockwise about the center at the origin. Where is the tip now? Word problem
- A spinner's pointer tip at (3, 0) rotates 180° about the origin. Where does it point now? Word problem
Tier 2 Intermediate One more step to think through
- Rotate the point (2, −3) 270° counterclockwise about the origin.
- Rotate triangle ABC with A(1, 2), B(3, 2) and C(1, 5) 90° counterclockwise about the origin.
- Point (2, 5) maps to (−2, −5), and point (−1, 3) maps to (1, −3). Which rotation about the origin is this?
- A Ferris wheel seat at (5, 0) turns 90° clockwise about the center at the origin. Where is the seat now? Word problem
- A game sprite has corners at (1, 1) and (2, 3). It rotates 90° counterclockwise about the origin. Where are the corners now? Word problem
- A marker at (−4, 1) on a turntable rotates 90° clockwise about the origin. Where is it now? Word problem
- A segment from (0, 0) to (5, 12) is rotated 90° about the origin. How long is the image? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- Rotate the point (6, 2) 90° counterclockwise about the point (2, 2).
- The point (3, 1) is rotated 90° counterclockwise about the origin, then 180° about the origin. Where does it end up?
- A door's hinge is at (1, 1), and its handle is at (4, 1). The door swings 90° counterclockwise about the hinge. Where is the handle now? Word problem
- A tile's corner moves from (2, 3) to (−3, 2) when the tile turns about the origin. Which rotation is this? Word problem
- A satellite dish marker at (−2, 5) rotates 270° counterclockwise about the origin. Where is it now? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- Rotate the point (5, 3) 90° clockwise about the point (1, 1). Above grade level
- A robot arm's tip at (3, 4) rotates 90° counterclockwise about the origin and is then reflected over the x-axis. Where does it end up? Word problem
- A figure is rotated 90° clockwise three times. What single rotation puts it in the same place? Word problem
Answers
Tier 1 Foundations
- (−4, −2). Apply (x, y) → (−x, −y) to each point.
- (−5, 1). Apply (x, y) → (−y, x) to each point.
- (−6, 0). Apply (x, y) → (−y, x) to each point.
- (4, 0). Apply (x, y) → (y, −x) to each point.
- (−3, 0). Apply (x, y) → (−x, −y) to each point.
Tier 2 Intermediate
- (−3, −2). Apply (x, y) → (y, −x) to each point.
- (−2, 1), (−2, 3), (−5, 1). Apply (x, y) → (−y, x) to each point.
- (x, y) → (−x, −y). Apply the rule to every original point; each one lands on its image point.
- (0, −5). Apply (x, y) → (y, −x) to each point.
- (−1, 1), (−3, 2). Apply (x, y) → (−y, x) to each point.
- (1, 4). Apply (x, y) → (y, −x) to each point.
- 13. Across 5, up or down 12: d² = 5² + 12² = 169, so d = 13.
Tier 3 Advanced
- (2, 6). Apply rotate 90° counterclockwise about (2, 2) to each point.
- (1, −3). Apply (x, y) → (−y, x); then (x, y) → (−x, −y) to each point.
- (1, 4). Apply rotate 90° counterclockwise about (1, 1) to each point.
- (x, y) → (−y, x). Apply the rule to every original point; each one lands on its image point.
- (5, 2). Apply (x, y) → (y, −x) to each point.
Tier 4 Challenge
- (3, −3). Apply rotate 90° clockwise about (1, 1) to each point.
- (−4, −3). Apply (x, y) → (−y, x); then (x, y) → (x, −y) to each point.
- 90° counterclockwise. Three clockwise quarter-turns make 270° clockwise, which lands in the same place as one quarter-turn counterclockwise.
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