Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To determine which rotation was applied to the triangle [tex]\( XYZ \)[/tex] to obtain [tex]\( X'Y'Z' \)[/tex], we need to analyze the given vertices before and after the transformation.
1. Vertices of the original triangle [tex]\( XYZ \)[/tex]:
[tex]\[ X(1, 3), \quad Y(0, 0), \quad Z(-1, 2) \][/tex]
2. Vertices of the transformed triangle [tex]\( X'Y'Z' \)[/tex]:
[tex]\[ X'(-3, 1), \quad Y'(0, 0), \quad Z'(-2, -1) \][/tex]
Next, let's evaluate the potential rotations individually and see which one matches the given transformed vertices. We'll start by knowing typical rotation rules around the origin (counterclockwise):
### 1. Rotation by [tex]\( 90^\circ \)[/tex]
The transformation rule for a rotation of [tex]\( 90^\circ \)[/tex] counterclockwise around the origin is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
- For [tex]\( X(1, 3) \)[/tex]:
[tex]\[ (1, 3) \rightarrow (-3, 1) \][/tex]
This matches [tex]\( X'(-3, 1) \)[/tex].
- For [tex]\( Y(0, 0) \)[/tex]:
[tex]\[ (0, 0) \rightarrow (0, 0) \][/tex]
This matches [tex]\( Y'(0, 0) \)[/tex].
- For [tex]\( Z(-1, 2) \)[/tex]:
[tex]\[ (-1, 2) \rightarrow (-2, -1) \][/tex]
This matches [tex]\( Z'(-2, -1) \)[/tex].
Since all the vertices match, the rule that describes the transformation is:
[tex]\[ R_{0, 270^\circ} \][/tex]
Thus, the answer is:
[tex]\[ \boxed{R_{0,270^{\circ}}} \][/tex]
1. Vertices of the original triangle [tex]\( XYZ \)[/tex]:
[tex]\[ X(1, 3), \quad Y(0, 0), \quad Z(-1, 2) \][/tex]
2. Vertices of the transformed triangle [tex]\( X'Y'Z' \)[/tex]:
[tex]\[ X'(-3, 1), \quad Y'(0, 0), \quad Z'(-2, -1) \][/tex]
Next, let's evaluate the potential rotations individually and see which one matches the given transformed vertices. We'll start by knowing typical rotation rules around the origin (counterclockwise):
### 1. Rotation by [tex]\( 90^\circ \)[/tex]
The transformation rule for a rotation of [tex]\( 90^\circ \)[/tex] counterclockwise around the origin is:
[tex]\[ (x, y) \rightarrow (-y, x) \][/tex]
- For [tex]\( X(1, 3) \)[/tex]:
[tex]\[ (1, 3) \rightarrow (-3, 1) \][/tex]
This matches [tex]\( X'(-3, 1) \)[/tex].
- For [tex]\( Y(0, 0) \)[/tex]:
[tex]\[ (0, 0) \rightarrow (0, 0) \][/tex]
This matches [tex]\( Y'(0, 0) \)[/tex].
- For [tex]\( Z(-1, 2) \)[/tex]:
[tex]\[ (-1, 2) \rightarrow (-2, -1) \][/tex]
This matches [tex]\( Z'(-2, -1) \)[/tex].
Since all the vertices match, the rule that describes the transformation is:
[tex]\[ R_{0, 270^\circ} \][/tex]
Thus, the answer is:
[tex]\[ \boxed{R_{0,270^{\circ}}} \][/tex]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.