Question: Complete the table below for the points $A$, $B$, $C$, and $D$ after they have been rotated $270^\circ$ anticlockwise about the origin.
Coordinates before
Coordinates after
$A$
$(2, 3)$
$( , )$
$B$
$(-2, 3)$
$( , )$
$C$
$(-2, -3)$
$( , )$
$D$
$(2, -3)$
$( , )$
1. **State the problem:** We need to find the coordinates of points $A$, $B$, $C$, and $D$ after a $270^\circ$ anticlockwise rotation about the origin.
2. **Formula for rotation:** Rotating a point $(x,y)$ by $\theta$ degrees anticlockwise about the origin results in new coordinates given by:
$$
(x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)
$$
3. **Apply for $270^\circ$ rotation:** Since $\cos 270^\circ = 0$ and $\sin 270^\circ = -1$, the formula simplifies to:
$$
(x', y') = (x \cdot 0 - y \cdot (-1), x \cdot (-1) + y \cdot 0) = (y, -x)
$$
4. **Calculate new coordinates:**
- For $A(2,3)$:
$$
(x', y') = (3, -2)
$$
- For $B(-2,3)$:
$$
(x', y') = (3, 2)
$$
- For $C(-2,-3)$:
$$
(x', y') = (-3, 2)
$$
- For $D(2,-3)$:
$$
(x', y') = (-3, -2)
$$
5. **Final answer:**
| Point | Before | After |
|-------|--------|-------|
| $A$ | $(2,3)$ | $(3,-2)$ |
| $B$ | $(-2,3)$ | $(3,2)$ |
| $C$ | $(-2,-3)$| $(-3,2)$ |
| $D$ | $(2,-3)$ | $(-3,-2)$|