Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Explore thousands of questions and answers from a knowledgeable community of experts ready to help you find solutions. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To determine which rotation transformation has been applied to triangle [tex]\( XYZ \)[/tex] resulting in [tex]\( X'Y'Z' \)[/tex], we'll analyze the coordinates of the vertices before and after the rotation.
Initial vertices of triangle [tex]\( XYZ \)[/tex]:
- [tex]\( X = (1, 3) \)[/tex]
- [tex]\( Y = (0, 0) \)[/tex]
- [tex]\( Z = (-1, 2) \)[/tex]
Vertices of the image triangle [tex]\( X'Y'Z' \)[/tex] after the rotation:
- [tex]\( X' = (-3, 1) \)[/tex]
- [tex]\( Y' = (0, 0) \)[/tex]
- [tex]\( Z' = (-2, -1) \)[/tex]
We need to check the coordinates of the vertices after applying different rotation transformations around the origin.
### 1. Rotation by [tex]\( 90^\circ \)[/tex] Counterclockwise ([tex]\( R_{0,90^\circ} \)[/tex])
The rule for rotating a point [tex]\((x, y)\)[/tex] by [tex]\( 90^\circ \)[/tex] counterclockwise around the origin is [tex]\((x, y) \to (-y, x)\)[/tex]:
- For [tex]\( X(1, 3) \)[/tex]:
[tex]\[ X \to (-3, 1) \][/tex]
- For [tex]\( Y(0, 0) \)[/tex]:
[tex]\[ Y \to (0, 0) \][/tex]
- For [tex]\( Z(-1, 2) \)[/tex]:
[tex]\[ Z \to (-2, -1) \][/tex]
These calculated points match exactly with the given vertices of [tex]\( X'Y'Z' \)[/tex]:
- [tex]\( X' = (-3, 1) \)[/tex]
- [tex]\( Y' = (0, 0) \)[/tex]
- [tex]\( Z' = (-2, -1) \)[/tex]
Thus, the rotation rule that describes the transformation from [tex]\( XYZ \)[/tex] to [tex]\( X'Y'Z' \)[/tex] is [tex]\( R_{0, 90^\circ} \)[/tex].
Therefore, the correct rule is:
[tex]\[ \boxed{R_{0,90^{\circ}}} \][/tex]
Initial vertices of triangle [tex]\( XYZ \)[/tex]:
- [tex]\( X = (1, 3) \)[/tex]
- [tex]\( Y = (0, 0) \)[/tex]
- [tex]\( Z = (-1, 2) \)[/tex]
Vertices of the image triangle [tex]\( X'Y'Z' \)[/tex] after the rotation:
- [tex]\( X' = (-3, 1) \)[/tex]
- [tex]\( Y' = (0, 0) \)[/tex]
- [tex]\( Z' = (-2, -1) \)[/tex]
We need to check the coordinates of the vertices after applying different rotation transformations around the origin.
### 1. Rotation by [tex]\( 90^\circ \)[/tex] Counterclockwise ([tex]\( R_{0,90^\circ} \)[/tex])
The rule for rotating a point [tex]\((x, y)\)[/tex] by [tex]\( 90^\circ \)[/tex] counterclockwise around the origin is [tex]\((x, y) \to (-y, x)\)[/tex]:
- For [tex]\( X(1, 3) \)[/tex]:
[tex]\[ X \to (-3, 1) \][/tex]
- For [tex]\( Y(0, 0) \)[/tex]:
[tex]\[ Y \to (0, 0) \][/tex]
- For [tex]\( Z(-1, 2) \)[/tex]:
[tex]\[ Z \to (-2, -1) \][/tex]
These calculated points match exactly with the given vertices of [tex]\( X'Y'Z' \)[/tex]:
- [tex]\( X' = (-3, 1) \)[/tex]
- [tex]\( Y' = (0, 0) \)[/tex]
- [tex]\( Z' = (-2, -1) \)[/tex]
Thus, the rotation rule that describes the transformation from [tex]\( XYZ \)[/tex] to [tex]\( X'Y'Z' \)[/tex] is [tex]\( R_{0, 90^\circ} \)[/tex].
Therefore, the correct rule is:
[tex]\[ \boxed{R_{0,90^{\circ}}} \][/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.