Answer :
Part A: Rotation by [tex]\( 270^\circ \)[/tex] Counterclockwise
To rotate a point by [tex]\( 270^\circ \)[/tex] counterclockwise about the origin, we use the transformation rules:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Let's apply this transformation to each vertex of triangle [tex]\( \triangle XYZ \)[/tex].
1. Vertex [tex]\( X(-4, -4) \)[/tex]:
- Apply the transformation:
[tex]\[ (-4, -4) \rightarrow (-4, 4) \][/tex]
2. Vertex [tex]\( Y(5, 5) \)[/tex]:
- Apply the transformation:
[tex]\[ (5, 5) \rightarrow (5, -5) \][/tex]
3. Vertex [tex]\( Z(3, -6) \)[/tex]:
- Apply the transformation:
[tex]\[ (3, -6) \rightarrow (-6, -3) \][/tex]
So, after rotating [tex]\( \triangle XYZ \)[/tex] by [tex]\( 270^\circ \)[/tex] counterclockwise, the new coordinates of the vertices are:
[tex]\[ X'(-4, 4), \ Y'(5, -5), \ Z'(-6, -3) \][/tex]
Part B: Rotation by [tex]\( 90^\circ \)[/tex] Clockwise
To rotate a point by [tex]\( 90^\circ \)[/tex] clockwise about the origin, we use a similar transformation rule:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Let's apply this transformation to each vertex of triangle [tex]\( \triangle XYZ \)[/tex].
1. Vertex [tex]\( X(-4, -4) \)[/tex]:
- Apply the transformation:
[tex]\[ (-4, -4) \rightarrow (-4, 4) \][/tex]
2. Vertex [tex]\( Y(5, 5) \)[/tex]:
- Apply the transformation:
[tex]\[ (5, 5) \rightarrow (5, -5) \][/tex]
3. Vertex [tex]\( Z(3, -6) \)[/tex]:
- Apply the transformation:
[tex]\[ (3, -6) \rightarrow (-6, -3) \][/tex]
So, after rotating [tex]\( \triangle XYZ \)[/tex] by [tex]\( 90^\circ \)[/tex] clockwise, the new coordinates of the vertices are:
[tex]\[ X''(-4, 4), \ Y''(5, -5), \ Z''(-6, -3) \][/tex]
Part C: Similarities and Differences Between the Two Rotations
Similarities:
- Both rotations result in the same final coordinates for the vertices.
- The transformation rule applied in both rotations is essentially the same, as the rule [tex]\( (x, y) \rightarrow (y, -x) \)[/tex] is used in both cases.
Differences:
- Conceptually, a [tex]\( 270^\circ \)[/tex] counterclockwise rotation is equivalent to a [tex]\( 90^\circ \)[/tex] clockwise rotation. The transformations seem different (counterclockwise vs. clockwise), but they result in the same rotation in the plane.
- The interpretation of directions is different: one is a rotation of 270 degrees counterclockwise, and the other is a rotation of 90 degrees clockwise.
In summary, despite the different conceptual directions, the resulting coordinates after these rotations in our particular case are the same for both transformations.
To rotate a point by [tex]\( 270^\circ \)[/tex] counterclockwise about the origin, we use the transformation rules:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Let's apply this transformation to each vertex of triangle [tex]\( \triangle XYZ \)[/tex].
1. Vertex [tex]\( X(-4, -4) \)[/tex]:
- Apply the transformation:
[tex]\[ (-4, -4) \rightarrow (-4, 4) \][/tex]
2. Vertex [tex]\( Y(5, 5) \)[/tex]:
- Apply the transformation:
[tex]\[ (5, 5) \rightarrow (5, -5) \][/tex]
3. Vertex [tex]\( Z(3, -6) \)[/tex]:
- Apply the transformation:
[tex]\[ (3, -6) \rightarrow (-6, -3) \][/tex]
So, after rotating [tex]\( \triangle XYZ \)[/tex] by [tex]\( 270^\circ \)[/tex] counterclockwise, the new coordinates of the vertices are:
[tex]\[ X'(-4, 4), \ Y'(5, -5), \ Z'(-6, -3) \][/tex]
Part B: Rotation by [tex]\( 90^\circ \)[/tex] Clockwise
To rotate a point by [tex]\( 90^\circ \)[/tex] clockwise about the origin, we use a similar transformation rule:
[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]
Let's apply this transformation to each vertex of triangle [tex]\( \triangle XYZ \)[/tex].
1. Vertex [tex]\( X(-4, -4) \)[/tex]:
- Apply the transformation:
[tex]\[ (-4, -4) \rightarrow (-4, 4) \][/tex]
2. Vertex [tex]\( Y(5, 5) \)[/tex]:
- Apply the transformation:
[tex]\[ (5, 5) \rightarrow (5, -5) \][/tex]
3. Vertex [tex]\( Z(3, -6) \)[/tex]:
- Apply the transformation:
[tex]\[ (3, -6) \rightarrow (-6, -3) \][/tex]
So, after rotating [tex]\( \triangle XYZ \)[/tex] by [tex]\( 90^\circ \)[/tex] clockwise, the new coordinates of the vertices are:
[tex]\[ X''(-4, 4), \ Y''(5, -5), \ Z''(-6, -3) \][/tex]
Part C: Similarities and Differences Between the Two Rotations
Similarities:
- Both rotations result in the same final coordinates for the vertices.
- The transformation rule applied in both rotations is essentially the same, as the rule [tex]\( (x, y) \rightarrow (y, -x) \)[/tex] is used in both cases.
Differences:
- Conceptually, a [tex]\( 270^\circ \)[/tex] counterclockwise rotation is equivalent to a [tex]\( 90^\circ \)[/tex] clockwise rotation. The transformations seem different (counterclockwise vs. clockwise), but they result in the same rotation in the plane.
- The interpretation of directions is different: one is a rotation of 270 degrees counterclockwise, and the other is a rotation of 90 degrees clockwise.
In summary, despite the different conceptual directions, the resulting coordinates after these rotations in our particular case are the same for both transformations.
