Rotation

Rotation: Turning Figures About a Center

Focus: Spin a figure around a fixed center point by a specified angle and direction.

Topics in This Lesson

🔑 Big Idea: A rotation is a rigid transformation specified by three pieces of information: a center, an angle, and a direction. It preserves distances, angles, and orientation.

What You'll Master

• Apply $90°,\, 180°,\, 270°$ rotation rules about the origin (both directions)
• Rotate a figure by rotating each vertex by the same angle
• Use translation + rotate + translate-back to rotate about any center
• Recognize that composing two rotations gives another rotation (or a translation)

Entrance Quiz: Rotation Readiness

🧭 What a Rotation Needs

To specify a rotation completely, you need:

1. A center $O$ (a single point — the pivot)
2. An angle $\theta$ (how far to turn)
3. A direction — counterclockwise (positive) or clockwise (negative)

Under a rotation about $O$ by angle $\theta$:

• $O \to O$ (the center is fixed)
• For any other point $P$, the image $P'$ lies on the circle centered at $O$ with radius $|OP|$, and $\angle POP' = \theta$.

💡 Convention: in math, positive angles are counterclockwise. So $90°$ alone means counterclockwise; "$90°$ clockwise" or "$-90°$" means the other direction.

📐 Coordinate Rules for Rotations About the Origin

How to remember these without memorizing

A counterclockwise $90°$ rotation sends the unit vectors as follows:

So $(x, y) = x\hat{\imath} + y\hat{\jmath} \to x(0, 1) + y(-1, 0) = (-y,\, x)$. Apply it again for $180°$, and so on.

🔑 Quick sanity check: rotate $(1, 0)$ counterclockwise by $90°$. The image should be $(0, 1)$, not $(0, -1)$. If your rule gives $(0, -1)$, you've used the clockwise rule.

✏️ Rotating a Whole Figure

Worked Example

Rotate triangle $A(1, 1),\; B(4, 1),\; C(2, 3)$ by $90°$ clockwise about the origin.

Rule: $(x, y) \to (y,\, -x)$.

Because rotations preserve orientation, $A' \to B' \to C'$ is traversed in the same direction as $A \to B \to C$. The triangle rotated rigidly into the fourth quadrant region.

🎯 Rotating About a Point Other Than the Origin

To rotate by angle $\theta$ about a center $C = (h, k)$:

1. Translate so that $C$ moves to the origin: $(x, y) \to (x - h,\, y - k)$.
2. Rotate about the origin using the rule for $\theta$.
3. Translate back: add $(h, k)$ to the result.

Worked Example

Rotate $(5, 3)$ by $90°$ counterclockwise about $(2, 1)$.

1. Shift: $(5 - 2,\, 3 - 1) = (3,\, 2)$.
2. Rotate $90°$ counterclockwise: $(3, 2) \to (-2,\, 3)$.
3. Shift back: $(-2 + 2,\, 3 + 1) = (0,\, 4)$.

So the image is $(0, 4)$.

Check: Rotations About the Origin

⚠️ Common Mistakes

1. Mixing up clockwise and counterclockwise rules. Use the anchor: $(1, 0)$ rotated $90°$ counterclockwise becomes $(0, 1)$. If your rule sends $(1, 0) \to (0, -1)$, you wrote the clockwise rule.

2. Negating only one coordinate for a $180°$ rotation. A $180°$ rotation negates both coordinates. Negating only one is a reflection over an axis.

3. Skipping the translate / translate-back steps when rotating about a non-origin point. The shortcut rules in the table only work about the origin.

4. Forgetting that the center is fixed. If a vertex lies at the center of rotation, its image is the vertex itself — don't move it.

Exit Quiz: Rotation Mastery