Question

Rectangle MNPQ is rotated using the origin as the center of rotation, resulting in rectangle M’N’P’Q’, as shown below.

On a coordinate plane, rectangle M N P Q has points (negative 2, 6), (0, 4), (negative 3, 1), (negative 5, 3). Rectangle M prime N prime P prime Q prime has points (6, 2), (4, 0), (1, 3), (3, 5).

Which rotation may have occurred?
a 45° rotation clockwise
a 45° rotation counterclockwise
a 90° rotation clockwise
a 90° rotation counterclockwise

Answer 1

To determine the type of rotation that transformed rectangle MNPQ to rectangle M'N'P'Q', we can analyze the coordinates of both rectangles.

**Original Rectangle MNPQ:**
- M: (-2, 6)
- N: (0, 4)
- P: (-3, 1)
- Q: (-5, 3)

**Transformed Rectangle M'N'P'Q':**
- M': (6, 2)
- N': (4, 0)
- P': (1, 3)
- Q': (3, 5)

### Step 1: Understanding the Rotation
Rotating a point around the origin (0,0) can be represented mathematically:
- A 90° clockwise rotation transforms a point \((x, y)\) to \((y, -x)\).
- A 90° counterclockwise rotation transforms a point \((x, y)\) to \((-y, x)\).
- A 45° clockwise rotation transforms a point \((x, y)\) to \(\left(\frac{x - y}{\sqrt{2}}, \frac{x + y}{\sqrt{2}}\right)\).
- A 45° counterclockwise rotation transforms a point \((x, y)\) to \(\left(\frac{x + y}{\sqrt{2}}, \frac{-x + y}{\sqrt{2}}\right)\).

### Step 2: Calculate Rotated Points
**Let's start with a 90° clockwise rotation:**
- For point M (-2, 6):
- \((-2, 6) \rightarrow (6, 2)\) (Matches M')

- For point N (0, 4):
- \((0, 4) \rightarrow (4, 0)\) (Matches N')

- For point P (-3, 1):
- \((-3, 1) \rightarrow (1, 3)\) (Matches P')

- For point Q (-5, 3):
- \((-5, 3) \rightarrow (3, 5)\) (Matches Q')

**Since all points from MNPQ to M'N'P'Q' match perfectly with the 90° clockwise transformation**, we can conclude that rectangle MNPQ was rotated 90° clockwise to become rectangle M'N'P'Q'.

### Conclusion
The correct answer is:
**a 90° rotation clockwise**.