Lesson 6-3: Explore Rotations

Property

A rotation is a rigid transformation that "turns" a figure around a fixed anchor point called the Center of Rotation. Because it is a rigid motion, the figure keeps its exact size and shape. To perfectly describe a rotation, you must have three pieces of information:

1. The Center: The fixed dot the shape spins around.
2. The Angle: How far it spins (e.g., $90^\circ$, $180^\circ$).
3. The Direction: Clockwise (CW, like a clock) or Counterclockwise (CCW, opposite of a clock).

Note: In mathematics, Counterclockwise (CCW) is always the standard, "positive" direction.

Examples

• Macro View: Think of a Ferris wheel. The center hub is the "Center of Rotation," and the passenger cars travel in circular paths around it. The cars don't change size as they spin.
• Micro Detail (Direction Equivalence): Spinning $90^\circ$ Clockwise lands you in the exact same spot as spinning $270^\circ$ Counterclockwise ($360^\circ - 90^\circ = 270^\circ$).
• Micro Detail (Distance): If point $A$ is 5 inches away from the center of rotation, its image $A'$ will also be exactly 5 inches away from the center.

Explanation

A common mistake is thinking the shape just rotates in place. Unless the center of rotation is inside the shape, the entire shape travels along an invisible circular track to a new location on the graph. The center point is the only thing in the entire universe that does not move during a rotation!

Property

Before using algebra, we construct rotations physically using two tools: a protractor (to get the exact angle and direction) and a compass or ruler (to keep the distance from the center perfectly equal).

Examples

• Rotating Point P $70^\circ$ CCW around Center C:
  1. Draw a straight guideline connecting center $C$ to point $P$.
  2. Place the protractor on $C$, align it with the guideline, and mark a $70^\circ$ angle in the Counterclockwise direction. Draw a new ray through that mark.
  3. Measure the exact distance from $C$ to $P$. Mark that same distance on the new ray. That spot is $P'$.
• Rotating a Triangle: To rotate triangle $DEF$, you must do the 3-step process above separately for corner $D$, then corner $E$, then corner $F$. Finally, connect the new dots to form $D'E'F'$.

Explanation

Why do we do this? Doing it by hand proves why a rotation is a rigid motion. The protractor guarantees the angle is right, and the compass guarantees the shape doesn't stretch or shrink. When you do this on paper, physically turn the paper with your hands—it helps your brain visualize where the final image should land!

Property

When the center of rotation is the origin $(0,0)$, we use algebraic rules to find the new coordinates. Assume all angles are Counterclockwise (CCW) unless stated otherwise.

• $90^\circ$ CCW: $(x, y) \rightarrow (-y, x)$[Swap the numbers, change the sign of the NEW first number]
• $180^\circ$: $(x, y) \rightarrow (-x, -y)$ [Do NOT swap numbers, just change BOTH signs]
• $270^\circ$ CCW (or $90^\circ$ CW): $(x, y) \rightarrow (y, -x)$ [Swap the numbers, change the sign of the NEW second number]

Examples

• $90^\circ$ CCW Rotation: Rotate $A(4, 5)$.
  • Step 1 (Swap): $(5, 4)$
  • Step 2 (Change first sign): $(-5, 4)$. So, $A'(-5, 4)$.
• $180^\circ$ Rotation: Rotate $B(-2, 7)$.
  • Keep the order, flip both signs: $(+2, -7)$. So, $B'(2, -7)$.
• $90^\circ$ CW (which is $270^\circ$ CCW): Rotate $C(-3, -6)$.
  • Step 1 (Swap): $(-6, -3)$
  • Step 2 (Change second sign): $(-6, +3)$. So, $C'(-6, 3)$.

Explanation

This is where the most errors happen! Students often try to swap the numbers and change the signs at the exact same time in their heads, which leads to messy mistakes with negatives. Always do it in two micro-steps: Write down the swapped numbers first, then apply the negative sign to the correct position. Also, remember that a $180^\circ$ rotation rule $(-x, -y)$ looks exactly like reflecting across both the x and y axes!