Question

The coordinates for a point and its image are Q(−3,−5)
and Q′(5,−3)
. Based on the mapping rules for rotations, which transformation occurred?(1 point)
Responses

a 270-degree (counterclockwise) rotation
a 270-degree (counterclockwise) rotation

a −180
-degree (clockwise) rotation or a 180-degree (counterclockwise) rotation
a negative 180 -degree (clockwise) rotation or a 180-degree (counterclockwise) rotation

a 90-degree (counterclockwise) rotation
a 90-degree (counterclockwise) rotation

a −90
-degree (clockwise) rotation

Answer 1

To determine the type of rotation transformation that occurred to map point Q(−3,−5) to point Q′(5,−3), we can analyze the coordinates through the standard rotation rules.

1. **Identify the original and image points:**
\( Q(-3, -5) \)
\( Q'(5, -3) \)

2. **Check for a 90-degree rotation:**
The transformation rule for a 90-degree counterclockwise rotation is:
\( (x, y) \rightarrow (-y, x) \)
Applying this to \( Q(-3, -5) \):
\( (-(-5), -3) = (5, -3) \)
This matches Q′, hence a 90-degree counterclockwise rotation seems to be appropriate.

3. **Check other angles of rotation for confirmation:**
- **270-degree counterclockwise rotation:** \( (x, y) \rightarrow (y, -x) \)
Applying to \( Q(-3, -5) \) gives \( (-5, 3) \), which does not match Q′.
- **180-degree rotation (clockwise or counterclockwise):** \( (x, y) \rightarrow (-x, -y) \)
Applying gives \( (3, 5) \), which does not match Q′.
- **90-degree clockwise rotation:** \( (x, y) \rightarrow (y, -x) \)
Applying gives \( (-5, 3) \), which does not match Q′.
- **270-degree clockwise rotation:** \( (x, y) \rightarrow (-y, x) \)
This would give \( (-(-5), -3) = (5, -3) \), which matches Q′.

4. Thus, we can confirm:
Both a 90-degree counterclockwise rotation and a 270-degree clockwise rotation result in the same transformation from Q to Q′.

In conclusion, the transformation that occurred was:

**a 90-degree (counterclockwise) rotation.**