What stays constant and what changes in a rotation — including rotations about a center other than the origin.
The Mnemonic
What a Rotation Preserves — and What It Doesn't
Rotation Rules covered the specific coordinate formulas for 90°/180°/270° rotations about the ORIGIN. This lesson covers rotations more generally — including about any center point — and focuses on what stays INVARIANT (unchanged) throughout any rotation.
What NEVER changes: every point's distance from the center of rotation, and the shape's size and angle measures (rotation is an isometry, so the image is always congruent to the pre-image). What DOES change: each point's angular position around the center, and the shape's overall orientation in the plane (though not whether it's "flipped" — unlike reflection, rotation never mirror-flips a shape).
Every point stays exactly the same distance from the center — only its angular position around that center changes
💡 Memory Trick
"What stays constant and what changes in a rotation." CONSTANT: distance from center, size, angle measures (it's an isometry). CHANGES: the point's position/angle around that center. Think of a clock hand: its length (distance from the center) never changes as it rotates — only which direction it's pointing does.
Why It Works
Rotations About Any Center, Not Just the Origin
Every point on a rotating shape traces out a circular arc centered at the ROTATION CENTER — which is why distance from that center never changes; a circle is, by definition, the set of points a FIXED distance from a center (straight back to the Circle Formulas and Equation of a Circle lessons). The center itself is the one point that never moves at all during a rotation.
The Rotation Rules formulas ((x,y)→(−y,x), etc.) only work because they're specifically centered at the origin — for a rotation about ANY other point, the same three-step process from Dilation Rules applies: subtract the center's coordinates, apply the origin-based rule, then add the center's coordinates back.
Using It In A Proof
Finding the Center and Angle of an Unknown Rotation
Given a pre-image and its rotated image (without being told the center or angle), you can reconstruct both using the invariant properties above.
1
Find the center of rotation
The center is EQUIDISTANT from each original point and its corresponding image point — meaning it lies on the perpendicular bisector of the segment connecting any point to its image. Find this for two different point-pairs; where the two perpendicular bisectors intersect is the center.
2
Find the angle of rotation
Once the center is known, measure the angle between a segment from the center to an original point and the segment from the center to that point's image.
3
Verify using a second point
Check that the SAME angle, measured from the SAME center, correctly maps a different point to its corresponding image — confirming the rotation was identified correctly.
Full Worked Example
Applying a Rotation About a Non-Origin Center
Given: Point P(5, 3) is rotated 90° counterclockwise about the center (2, 3), instead of the origin. Find: the image.
1
Subtract the center's coordinates
(5−2, 3−3) = (3, 0) — this treats the center as a temporary origin.
2
Apply the origin-based 90° rule
(x,y)→(−y,x), so (3,0) → (0, 3).
3
Add the center's coordinates back
(0+2, 3+3) = (2, 6).
4
Verify the distance-from-center invariant
Distance from center (2,3) to original P(5,3): √[(5−2)²+(3−3)²] = √9 = 3. Distance from center to image (2,6): √[(2−2)²+(6−3)²] = √9 = 3. ✓ Equal, confirming the invariant held.
This distance check is a fast, reliable way to catch an error in an off-origin rotation calculation, since the distance from the CENTER (not the origin) must stay exactly the same.
🎯 Quick Worked Example
A shape is rotated about some center, and a point 6 units from that center maps to an image point. How far should the image point be from the same center?
1
Recall the invariant property. Distance from the center of rotation never changes.
2
Apply it directly. Since the original point was 6 units from the center, the image point must also be exactly 6 units from that same center.
3
Conclude. The image point is 6 units from the center — no calculation needed beyond recognizing this invariant.
📌 Exam Application
Problems that ask you to FIND the center or angle of an unknown rotation (rather than apply a known one) specifically test the perpendicular bisector method — this connects directly back to the Perpendicular Bisector lesson from Coordinate Geometry, since the center of rotation is found using exactly that same construction.
⚠️ Most Common Properties of Rotations Mistakes
Trap 1 — Assuming the Rotation Rules formulas work for any center: (x,y)→(−y,x) and similar formulas are ORIGIN-ONLY — a rotation about a different center requires the subtract-rotate-add-back process shown here.
Trap 2 — Forgetting that distance from center (not from the origin) is the true invariant: If the rotation center isn't the origin, distance from the ORIGIN can absolutely change during the rotation — only distance from the actual rotation CENTER stays fixed.
✓ Quick Self-Test
1) What measurement stays invariant for every point during a rotation? 2) Why can't the origin-based Rotation Rules formulas be used directly for an off-origin rotation? 3) What construction is used to find the center of an unknown rotation? 4) Point Q(6,2) is rotated 90° CCW about center (1,2) — find its image using the subtract-rotate-add-back method. 5) Why is checking distance from the rotation center a reliable way to verify a rotation calculation?