To find the transformation that maps the original square with vertices \((3,3)\), \((6,3)\), \((6,6)\), and \((3,6)\) to the new square with vertices \((-3,-3)\), \((-6,-3)\), \((-6,-6)\), and \((-3,-6)\), we can analyze the coordinates.
- The original square's center is at \((4.5, 4.5)\).
- The new square's center is at \((-4.5, -4.5)\).
The transformation from one square's center to the other's suggests some form of rotation or reflection rather than translation, since one is in the positive quadrant and the other in the negative quadrant.
To determine if a rotation could apply, we can examine the possibility of a \(180\) degree rotation. Rotating a point \((x, y)\) \(180\) degrees about the origin results in the point \((-x, -y)\):
- For the vertex \((3, 3)\):
- Rotate \(180^\circ\) to get \((-3, -3)\)
- For the vertex \((6, 3)\):
- Rotate \(180^\circ\) to get \((-6, -3)\)
- For the vertex \((6, 6)\):
- Rotate \(180^\circ\) to get \((-6, -6)\)
- For the vertex \((3, 6)\):
- Rotate \(180^\circ\) to get \((-3, -6)\)
Since each vertex of the original square transforms perfectly into the corresponding vertex on the new square under a \(180\) degree rotation about the origin, the transformation applied is indeed:
Rotation \(180\) degrees about the origin.