Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Experience the convenience of getting reliable answers to your questions from a vast network of knowledgeable experts. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
