Question

Figure FGH has vertices located at F(1, 3), G(–1, 2), and H(2, 1). The figure is rotated using the origin as the center of rotation. The image has vertices located at F’(3, –1), G’(2, 1), and H’(1, –2). Which rotation could have taken place?
a 45° clockwise rotation
a 90° clockwise rotation
a 90° counterclockwise rotation
a 180° counterclockwise rotation

Answer 1

To determine which rotation took place for the vertices of figure FGH when rotated around the origin, we can analyze the original and image coordinates.

**Original vertices:**
- F(1, 3)
- G(–1, 2)
- H(2, 1)

**Image vertices after rotation:**
- F’(3, –1)
- G’(2, 1)
- H’(1, –2)

We will apply the rotation transformations for each option to check which one matches the transformation that occurred.

1. **90° clockwise rotation:**
- The transformation rule is (x, y) → (y, -x).
- Applying to F(1, 3):
\((1, 3) \rightarrow (3, -1)\) (matches F’)
- Applying to G(–1, 2):
\((-1, 2) \rightarrow (2, 1)\) (matches G’)
- Applying to H(2, 1):
\((2, 1) \rightarrow (1, -2)\) (does not match H’)

**Partial match, proceed to check other rotations.**

2. **90° counterclockwise rotation:**
- The transformation rule is (x, y) → (-y, x).
- Applying to F(1, 3):
\((1, 3) \rightarrow (-3, 1)\) (does not match F’)

**Not a match.**

3. **180° counterclockwise rotation:**
- The transformation rule is (x, y) → (-x, -y).
- Applying to F(1, 3):
\((1, 3) \rightarrow (-1, -3)\) (does not match F’)

**Not a match.**

4. **45° clockwise rotation:**
- The rules for a 45° rotation are more complex and typically involve using sine and cosine (specifically \(x' = x \cos(45°) + y \sin(45°)\) and \(y' = -x \sin(45°) + y \cos(45°)\)).
- Calculating this would be quite complex and typically does not yield simple integer coordinates.

Given that the only transformation that matched two points (F and G) and partially matched the third point was the 90° clockwise rotation.

Thus, the rotation that took place is **a 90° clockwise rotation**.