Coordinate transformation rules for standard rotations about the origin.
The Mnemonic
Three Standard Rotation Angles, Three Rules
Rotating a point (x, y) counterclockwise about the origin by three standard angles: 90°: (x,y) → (−y, x). 180°: (x,y) → (−x, −y). 270° counterclockwise (equivalent to 90° clockwise): (x,y) → (y, −x).
Notice 180° is really just applying the 90° rule TWICE — rotating another 90° from an already-90°-rotated point. And 270° is applying it three times total.
Rotating counterclockwise about the origin by 90° swaps and flips one coordinate; 180° flips both signs
💡 Memory Trick
"Coordinate transformation rules for standard rotations." 90°: (x,y)→(−y,x) — swap and negate the NEW first coordinate. 180°: (x,y)→(−x,−y) — negate BOTH (this is also just two reflections' worth of sign flips combined). 270°: (x,y)→(y,−x) — swap and negate the NEW second coordinate instead. Each one builds on the last by applying another 90° turn.
Why It Works
Building 180° and 270° From the 90° Rule
Since a 180° rotation is exactly two consecutive 90° rotations, you can derive its rule by applying the 90° rule twice in a row. Start with (x, y); one 90° turn gives (−y, x). Apply the SAME rule again to this new point — treating −y as the new "x" and x as the new "y" — which gives (−x, −y). This composition (chaining transformations) previews exactly what the upcoming Composition of Transformations lesson covers in depth.
270° similarly comes from applying the 90° rule a third time: (−x,−y) → (y,−x). Each additional 90° rotation is just re-applying the same swap-and-negate pattern to whatever the current coordinates are, not a brand new independent rule to memorize from scratch.
Using It In A Proof
Applying the Correct Rotation Rule
Since all rotations here are about the ORIGIN specifically, confirming that detail — plus correctly matching the angle to its rule — is the main skill.
1
Confirm rotation is about the origin
These specific rules only work for rotations centered at (0,0) — a rotation about a different center point requires additional steps (translating to the origin first, rotating, then translating back).
2
Identify the exact angle and direction
Match the given angle (90°, 180°, or 270°) to its rule — double check whether counterclockwise is specified, since that's the standard default direction.
3
Apply the rule to each coordinate
Substitute the point's x and y values into the matched rule.
Full Worked Example
Rotating a Triangle 180°
Given: Triangle with vertices A(3, 2), B(6, 2), C(3, 5). Find: the image after rotating 180° about the origin.
The original triangle sits entirely in Quadrant I (all positive coordinates); after a 180° rotation, the image sits entirely in Quadrant III (all negative) — exactly the diagonally opposite quadrant, which makes sense for a half-turn.
This quadrant check (from Coordinate Plane Quadrants) is a fast way to verify a 180° rotation landed correctly — it should always move a shape to the exact opposite quadrant.
🎯 Quick Worked Example
Rotate the point (4, −1) by 90° counterclockwise about the origin.
1
Identify the rule. 90° CCW: (x,y) → (−y,x).
2
Apply the rule. Original point (4,−1): new x = −(−1) = 1, new y = 4.
3
State the image. The rotated point is (1, 4).
📌 Exam Application
Watch for problems specifying CLOCKWISE rotation instead of the standard counterclockwise default — a 90° clockwise rotation actually uses the same rule as a 270° counterclockwise rotation, (x,y)→(y,−x), since they land in the same final position; always confirm which direction is specified before picking a rule.
⚠️ Most Common Rotation Rules Mistakes
Trap 1 — Mixing up the 90° and 270° rules: (x,y)→(−y,x) for 90° and (x,y)→(y,−x) for 270° look similar but have their negative signs on opposite coordinates — applying the wrong one rotates the point a full 180° away from where it should actually land.
Trap 2 — Forgetting these rules only apply to rotation about the origin: A rotation about any other center point requires a different (multi-step) process — these three clean rules are specifically an origin-only shortcut.
✓ Quick Self-Test
1) State the rule for a 90° counterclockwise rotation about the origin. 2) State the rule for a 180° rotation. 3) State the rule for a 270° counterclockwise rotation. 4) Rotate the point (−2, 6) by 180° about the origin. 5) Why does a 180° rotation always move a shape to the diagonally opposite quadrant?