To determine the correct rotation rule, we need to examine how each point is transformed through rotation.
Let's start with point [tex]\( A \)[/tex]. Given:
[tex]\[ A = (-3, 4) \][/tex]
[tex]\[ A' = (4, 3) \][/tex]
We notice that one common property of rotation transformations is the relationship between the original point and its image. Specifically, we can recognize the behavior of 90-degree increments in rotations around the origin.
Let's explore the general rotation rules around the origin:
1. 90 Degrees Clockwise:
- Formula: [tex]\( (x, y) \to (y, -x) \)[/tex]
2. 180 Degrees:
- Formula: [tex]\( (x, y) \to (-x, -y) \)[/tex]
3. 270 Degrees Clockwise (or 90 Degrees Counterclockwise):
- Formula: [tex]\( (x, y) \to (-y, x) \)[/tex]
4. 360 Degrees:
- Formula: (x, y) \to (x, y) (no change)
We will apply these transformations to point [tex]\( A \)[/tex] and see which matches [tex]\( A' \)[/tex].
### Applying the Rotation Rules
1. 90 Degrees Clockwise:
- Applying: [tex]\( (-3, 4) \to (4, -(-3)) = (4, 3) \)[/tex]
- This matches [tex]\( A' = (4, 3) \)[/tex].
2. 180 Degrees:
- Applying: [tex]\( (-3, 4) \to (-( -3), -4) = (3, -4) \)[/tex]
- This does not match [tex]\( A' = (4, 3) \)[/tex].
3. 270 Degrees Clockwise (or 90 Degrees Counterclockwise):
- Applying: [tex]\( (-3, 4) \to (-4, 3) \)[/tex]
- This does not match [tex]\( A' = (4, 3) \)[/tex].
4. 360 Degrees:
- Applying: [tex]\( (-3, 4) \to (-3, 4) \)[/tex]
- This does not match [tex]\( A' = (4, 3) \)[/tex].
By this result, it looks like a 90-degree clockwise rotation maps [tex]\( A \)[/tex] to [tex]\( A' \)[/tex]. To ensure this rule applies consistently, we should verify it with points [tex]\( B \)[/tex] and [tex]\( C \)[/tex].
### Checking for Point [tex]\( B \)[/tex]:
Given:
[tex]\[ B = (4, -5) \][/tex]
[tex]\[ B' = (-5, -4) \][/tex]
1. 90 Degrees Clockwise:
- Applying: [tex]\( (4, -5) \to (-5, -(4)) = (-5, -4) \)[/tex]
- This matches [tex]\( B' = (-5, -4) \)[/tex].
### Checking for Point [tex]\( C \)[/tex]:
Given:
[tex]\[ C = (1, 6) \][/tex]
[tex]\[ C' = (6, -1) \][/tex]
1. 90 Degrees Clockwise:
- Applying: [tex]\( (1, 6) \to (6, -1) = (6, -1) \)[/tex]
- This matches [tex]\( C' = (6, -1) \)[/tex].
Since the transformation for all points [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] align with the 90-degree clockwise rotation rule, we can conclusively determine:
The rule describing the rotation is [tex]\( R_{0,90^{\circ}} \)[/tex].
