🎯⭐ INTERACTIVE LESSON

Rotation

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Rotation - Complete Interactive Lesson

Part 1: 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

Section
What a Rotation Needs
Coordinate Rules for Rotations About the Origin
Rotating a Whole Figure
Rotations About Points Other Than the Origin
Composing Two Rotations

🔑 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:

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

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

$O \to O$ (the center is fixed)

📐 Coordinate Rules for Rotations About the Origin

Rotation (about origin)	Coordinate rule
$90°$ counterclockwise	$(x, y) \to (-y,\, x)$
$180°$ (either direction)

✏️ Rotating a Whole Figure

Worked Example

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

Rule: .

🎯 Rotating About a Point Other Than the Origin

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

Translate so that $C$ moves to the origin: $(x, y) \to (x - h, y - k)$ .

Check: Rotations About the Origin

⚠️ Common Mistakes

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.

Exit Quiz: Rotation Mastery

For any other point

P

, the image

P'

lies on the circle centered at

O

with radius

|OP|

, and

\angle POP' = \theta

|OP'| \;=\; |OP|, \qquad \angle POP' \;=\; \theta.

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

How to remember these without memorizing

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

\hat{\imath} = (1, 0) \;\to\; (0, 1) = \hat{\jmath}, \qquad \hat{\jmath} = (0, 1) \;\to\; (-1, 0) = -\hat{\imath}.

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.

(x, y) \to (y,\, -x)

(x, y) \to (y, - x)

Vertex	Rule applied	Image
$A(1, 1)$	$(1,\, -1)$	$A'(1, -1)$
$B(4, 1)$	$(1,\, -4)$	$B'(1, -4)$
$C(2, 3)$	$(3,\, -2)$	$C'(3, -2)$

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.

(x, y) \to (x - h, y - k)

Rotate about the origin using the rule for

\theta

Translate back: add

(h, k)

to the result.

\text{Rotate about } (h, k) \;=\; T_{(h,k)} \circ R_\theta \circ T_{(-h,\,-k)}

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

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

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

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

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

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.