Answer :
To determine the rule describing the given rotation, we need to examine the given points and their images after rotation.
### Step-by-step Solution:
1. Identify the coordinates:
- [tex]\( A(-3, 4) \)[/tex] maps to [tex]\( A' (4, 3) \)[/tex]
- [tex]\( B(4, -5) \)[/tex] maps to [tex]\( B' (-5, -4) \)[/tex]
- [tex]\( C(1, 6) \)[/tex] maps to [tex]\( C' (6, -1) \)[/tex]
2. Consider the standard rotation rules around the origin [tex]\( (0,0) \)[/tex]:
- [tex]\( R_{0, 90^{\circ}}: (x, y) \rightarrow (-y, x) \)[/tex]
- [tex]\( R_{0, 180^{\circ}}: (x, y) \rightarrow (-x, -y) \)[/tex]
- [tex]\( R_{0, 270^{\circ}}: (x, y) \rightarrow (y, -x) \)[/tex]
- [tex]\( R_{0, 360^{\circ}}: (x, y) \rightarrow (x, y) \)[/tex]
3. Check each rotation:
- For [tex]\( R_{0, 90^{\circ}} \)[/tex]:
- For [tex]\( A(-3, 4) \)[/tex], applying [tex]\( (-y, x) \)[/tex]:
[tex]\[ (-4, -3) \neq (4, 3) \][/tex]
- [tex]\( R_{0, 90^{\circ}} \)[/tex] does not fit because [tex]\( A \neq A' \)[/tex].
- For [tex]\( R_{0, 180^{\circ}} \)[/tex]:
- For [tex]\( A(-3, 4) \)[/tex], applying [tex]\( (-x, -y) \)[/tex]:
[tex]\[ (3, -4) \neq (4, 3) \][/tex]
- [tex]\( R_{0, 180^{\circ}} \)[/tex] does not fit because [tex]\( A \neq A' \)[/tex].
- For [tex]\( R_{0, 270^{\circ}} \)[/tex]:
- For [tex]\( A(-3, 4) \)[/tex], applying [tex]\( (y, -x) \)[/tex]:
[tex]\[ (4, 3) = (4, 3) \][/tex]
Remarkably, [tex]\( A = A' \)[/tex].
- For [tex]\( B(4, -5) \)[/tex], applying [tex]\( (y, -x) \)[/tex]:
[tex]\[ (-5, -4) = (-5, -4) \][/tex]
Similarly, [tex]\( B = B' \)[/tex].
- For [tex]\( C(1, 6) \)[/tex], applying [tex]\( (y, -x) \)[/tex]:
[tex]\[ (6, -1) = (6, -1) \][/tex]
Lastly, [tex]\( C = C' \)[/tex].
Thus, [tex]\( R_{0, 270^{\circ}} \)[/tex] fits all points correctly.
- For [tex]\( R_{0, 360^{\circ}} \)[/tex]:
- For [tex]\( A(-3, 4) \)[/tex], applying [tex]\( (x, y) \)[/tex]:
[tex]\[ (-3, 4) \neq (4, 3) \][/tex]
- [tex]\( R_{0, 360^{\circ}} \)[/tex] does not fit because [tex]\( A \neq A' \)[/tex].
4. Conclusion:
The rotation that maps [tex]\( A(-3, 4) \)[/tex] to [tex]\( A' (4, 3) \)[/tex], [tex]\( B(4, -5) \)[/tex] to [tex]\( B' (-5, -4) \)[/tex], and [tex]\( C(1, 6) \)[/tex] to [tex]\( C' (6, -1) \)[/tex] is:
[tex]\[ \boxed{R_{0, 270^{\circ}}} \][/tex]
### Step-by-step Solution:
1. Identify the coordinates:
- [tex]\( A(-3, 4) \)[/tex] maps to [tex]\( A' (4, 3) \)[/tex]
- [tex]\( B(4, -5) \)[/tex] maps to [tex]\( B' (-5, -4) \)[/tex]
- [tex]\( C(1, 6) \)[/tex] maps to [tex]\( C' (6, -1) \)[/tex]
2. Consider the standard rotation rules around the origin [tex]\( (0,0) \)[/tex]:
- [tex]\( R_{0, 90^{\circ}}: (x, y) \rightarrow (-y, x) \)[/tex]
- [tex]\( R_{0, 180^{\circ}}: (x, y) \rightarrow (-x, -y) \)[/tex]
- [tex]\( R_{0, 270^{\circ}}: (x, y) \rightarrow (y, -x) \)[/tex]
- [tex]\( R_{0, 360^{\circ}}: (x, y) \rightarrow (x, y) \)[/tex]
3. Check each rotation:
- For [tex]\( R_{0, 90^{\circ}} \)[/tex]:
- For [tex]\( A(-3, 4) \)[/tex], applying [tex]\( (-y, x) \)[/tex]:
[tex]\[ (-4, -3) \neq (4, 3) \][/tex]
- [tex]\( R_{0, 90^{\circ}} \)[/tex] does not fit because [tex]\( A \neq A' \)[/tex].
- For [tex]\( R_{0, 180^{\circ}} \)[/tex]:
- For [tex]\( A(-3, 4) \)[/tex], applying [tex]\( (-x, -y) \)[/tex]:
[tex]\[ (3, -4) \neq (4, 3) \][/tex]
- [tex]\( R_{0, 180^{\circ}} \)[/tex] does not fit because [tex]\( A \neq A' \)[/tex].
- For [tex]\( R_{0, 270^{\circ}} \)[/tex]:
- For [tex]\( A(-3, 4) \)[/tex], applying [tex]\( (y, -x) \)[/tex]:
[tex]\[ (4, 3) = (4, 3) \][/tex]
Remarkably, [tex]\( A = A' \)[/tex].
- For [tex]\( B(4, -5) \)[/tex], applying [tex]\( (y, -x) \)[/tex]:
[tex]\[ (-5, -4) = (-5, -4) \][/tex]
Similarly, [tex]\( B = B' \)[/tex].
- For [tex]\( C(1, 6) \)[/tex], applying [tex]\( (y, -x) \)[/tex]:
[tex]\[ (6, -1) = (6, -1) \][/tex]
Lastly, [tex]\( C = C' \)[/tex].
Thus, [tex]\( R_{0, 270^{\circ}} \)[/tex] fits all points correctly.
- For [tex]\( R_{0, 360^{\circ}} \)[/tex]:
- For [tex]\( A(-3, 4) \)[/tex], applying [tex]\( (x, y) \)[/tex]:
[tex]\[ (-3, 4) \neq (4, 3) \][/tex]
- [tex]\( R_{0, 360^{\circ}} \)[/tex] does not fit because [tex]\( A \neq A' \)[/tex].
4. Conclusion:
The rotation that maps [tex]\( A(-3, 4) \)[/tex] to [tex]\( A' (4, 3) \)[/tex], [tex]\( B(4, -5) \)[/tex] to [tex]\( B' (-5, -4) \)[/tex], and [tex]\( C(1, 6) \)[/tex] to [tex]\( C' (6, -1) \)[/tex] is:
[tex]\[ \boxed{R_{0, 270^{\circ}}} \][/tex]
