Question

Describe the transformation in the diagram.
Point a(-4,-2), point b(-3,-1), c(-1,-2)
Point A'(2,-4), Point b'(1,-3), Point c'(2,-1)

90 degree clockwise rotation about the origin.
90 degree counterclockwise rotation about the origin.
270 degree counterclockwise rotation about the origin.
180 degree counterclockwise rotation about the origin

Answer 1

To analyze the transformation between the original points and their corresponding transformed points, let's look at the coordinates of each set of points:

Original Points:
- Point a(-4, -2)
- Point b(-3, -1)
- Point c(-1, -2)

Transformed Points:
- Point A'(2, -4)
- Point b'(1, -3)
- Point c'(2, -1)

Now, let's go through the types of transformations to determine which one applies:

### 1. **90-degree Clockwise Rotation:**
Using the formula for a 90-degree clockwise rotation about the origin, where \((x, y)\) transforms to \((y, -x)\):
- For a: \( (-4, -2) \to (-2, 4) \) (not matching A')
- For b: \( (-3, -1) \to (-1, 3) \) (not matching b')
- For c: \( (-1, -2) \to (-2, 1) \) (not matching c')

This transformation does not match.

### 2. **90-degree Counterclockwise Rotation:**
Using the formula for a 90-degree counterclockwise rotation about the origin, where \((x, y)\) transforms to \((-y, x)\):
- For a: \( (-4, -2) \to (2, -4) \) (matches A')
- For b: \( (-3, -1) \to (1, -3) \) (matches b')
- For c: \( (-1, -2) \to (2, -1) \) (matches c')

This transformation matches all points.

### 3. **270-degree Counterclockwise Rotation:**
This is equivalent to a 90-degree clockwise rotation. Since we already verified that the transformation is not a 90-degree clockwise rotation, this transformation also will not match.

### 4. **180-degree Counterclockwise Rotation:**
Using the formula for a 180-degree rotation about the origin, \((x, y)\) transforms to \((-x, -y)\):
- For a: \( (-4, -2) \to (4, 2) \) (not matching A')
- For b: \( (-3, -1) \to (3, 1) \) (not matching b')
- For c: \( (-1, -2) \to (1, 2) \) (not matching c')

This transformation does not match.

### Conclusion:
The transformation that applies to the points is a **90-degree counterclockwise rotation about the origin.**