Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
Let's solve this step-by-step.
We have the original vertices of the triangle:
- [tex]\( B(-3,0) \)[/tex]
- [tex]\( C(2,-1) \)[/tex]
- [tex]\( D(-1,2) \)[/tex]
And we want to transform these vertices to:
- [tex]\( B^{\circ}(1,-2) \)[/tex]
- [tex]\( C^{-}(0,3) \)[/tex]
- [tex]\( D^(3,0) \)[/tex]
Let's analyze each possible transformation to see which one works.
### Option 1: [tex]\((x, y) \rightarrow (x+1, y+1) \rightarrow (y, x)\)[/tex]
1. First transformation: [tex]\((x, y) \rightarrow (x+1, y+1)\)[/tex]
- [tex]\( B(-3,0) \rightarrow (-3+1, 0+1) = (-2, 1) \)[/tex]
- [tex]\( C(2,-1) \rightarrow (2+1, -1+1) = (3, 0) \)[/tex]
- [tex]\( D(-1,2) \rightarrow (-1+1, 2+1) = (0, 3) \)[/tex]
2. Second transformation: [tex]\((x+1, y+1) \rightarrow (y, x)\)[/tex]
- [tex]\( B(-2,1) \rightarrow (1, -2) \)[/tex]
- [tex]\( C(3,0) \rightarrow (0, 3) \)[/tex]
- [tex]\( D(0,3) \rightarrow (3, 0) \)[/tex]
This transformation matches our target vertices exactly:
- [tex]\( B^{\circ}(1,-2) \)[/tex]
- [tex]\( C^{-}(0,3) \)[/tex]
- [tex]\( D^(3,0) \)[/tex]
Therefore, Option 1 is indeed the correct transformation.
### Option 2: [tex]\((x, y) \rightarrow (x+1, y+1) \rightarrow (-x, y)\)[/tex]
1. First transformation: [tex]\((x, y) \rightarrow (x+1, y+1)\)[/tex]
- [tex]\( B(-3,0) \rightarrow (-3+1, 0+1) = (-2, 1) \)[/tex]
- [tex]\( C(2,-1) \rightarrow (2+1, -1+1) = (3, 0) \)[/tex]
- [tex]\( D(-1,2) \rightarrow (-1+1, 2+1) = (0, 3) \)[/tex]
2. Second transformation: [tex]\((x+1, y+1) \rightarrow (-x, y)\)[/tex]
- [tex]\( B(-2,1) \rightarrow (2, 1) \)[/tex]
- [tex]\( C(3,0) \rightarrow (-3, 0) \)[/tex]
- [tex]\( D(0,3) \rightarrow (0, 3) \)[/tex]
These vertices do not match our target vertices. Thus, Option 2 is not correct.
### Option 3: [tex]\((x, y) \rightarrow (x,-y) \rightarrow (x+2, y+2)\)[/tex]
1. First transformation: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- [tex]\( B(-3,0) \rightarrow (-3, 0) \)[/tex]
- [tex]\( C(2,-1) \rightarrow (2, 1) \)[/tex]
- [tex]\( D(-1,2) \rightarrow (-1, -2) \)[/tex]
2. Second transformation: [tex]\((x, -y) \rightarrow (x+2, y+2)\)[/tex]
- [tex]\( B(-3,0) \rightarrow (-3+2, 0+2) = (-1, 2) \)[/tex]
- [tex]\( C(2,1) \rightarrow (2+2, 1+2) = (4, 3) \)[/tex]
- [tex]\( D(-1,-2) \rightarrow (-1+2, -2+2) = (1, 0) \)[/tex]
These vertices do not match our target vertices. Thus, Option 3 is not correct.
### Option 4: [tex]\((x, y) \rightarrow (-x, y) \rightarrow (x+2, y+2)\)[/tex]
1. First transformation: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- [tex]\( B(-3, 0) \rightarrow (3, 0) \)[/tex]
- [tex]\( C(2, -1) \rightarrow (-2, -1) \)[/tex]
- [tex]\( D(-1, 2) \rightarrow (1, 2) \)[/tex]
2. Second transformation: [tex]\(( - x, y) \rightarrow (x+2, y+2)\)[/tex]
- [tex]\( B(3, 0) \rightarrow (3+2, 0+2) = (5, 2) \)[/tex]
- [tex]\( C(-2, -1) \rightarrow (-2+2, -1+2) = (0, 1) \)[/tex]
- [tex]\( D(1, 2) \rightarrow (1+2, 2+2) = (3, 4) \)[/tex]
These vertices do not match our target vertices. Thus, Option 4 is not correct.
### Conclusion
Among all the given options, only Option 1 results in the desired transformed vertices. Therefore, the correct transformation is:
[tex]\[ \boxed{(x, y) \rightarrow (x+1, y+1) \rightarrow (y, x)} \][/tex]
We have the original vertices of the triangle:
- [tex]\( B(-3,0) \)[/tex]
- [tex]\( C(2,-1) \)[/tex]
- [tex]\( D(-1,2) \)[/tex]
And we want to transform these vertices to:
- [tex]\( B^{\circ}(1,-2) \)[/tex]
- [tex]\( C^{-}(0,3) \)[/tex]
- [tex]\( D^(3,0) \)[/tex]
Let's analyze each possible transformation to see which one works.
### Option 1: [tex]\((x, y) \rightarrow (x+1, y+1) \rightarrow (y, x)\)[/tex]
1. First transformation: [tex]\((x, y) \rightarrow (x+1, y+1)\)[/tex]
- [tex]\( B(-3,0) \rightarrow (-3+1, 0+1) = (-2, 1) \)[/tex]
- [tex]\( C(2,-1) \rightarrow (2+1, -1+1) = (3, 0) \)[/tex]
- [tex]\( D(-1,2) \rightarrow (-1+1, 2+1) = (0, 3) \)[/tex]
2. Second transformation: [tex]\((x+1, y+1) \rightarrow (y, x)\)[/tex]
- [tex]\( B(-2,1) \rightarrow (1, -2) \)[/tex]
- [tex]\( C(3,0) \rightarrow (0, 3) \)[/tex]
- [tex]\( D(0,3) \rightarrow (3, 0) \)[/tex]
This transformation matches our target vertices exactly:
- [tex]\( B^{\circ}(1,-2) \)[/tex]
- [tex]\( C^{-}(0,3) \)[/tex]
- [tex]\( D^(3,0) \)[/tex]
Therefore, Option 1 is indeed the correct transformation.
### Option 2: [tex]\((x, y) \rightarrow (x+1, y+1) \rightarrow (-x, y)\)[/tex]
1. First transformation: [tex]\((x, y) \rightarrow (x+1, y+1)\)[/tex]
- [tex]\( B(-3,0) \rightarrow (-3+1, 0+1) = (-2, 1) \)[/tex]
- [tex]\( C(2,-1) \rightarrow (2+1, -1+1) = (3, 0) \)[/tex]
- [tex]\( D(-1,2) \rightarrow (-1+1, 2+1) = (0, 3) \)[/tex]
2. Second transformation: [tex]\((x+1, y+1) \rightarrow (-x, y)\)[/tex]
- [tex]\( B(-2,1) \rightarrow (2, 1) \)[/tex]
- [tex]\( C(3,0) \rightarrow (-3, 0) \)[/tex]
- [tex]\( D(0,3) \rightarrow (0, 3) \)[/tex]
These vertices do not match our target vertices. Thus, Option 2 is not correct.
### Option 3: [tex]\((x, y) \rightarrow (x,-y) \rightarrow (x+2, y+2)\)[/tex]
1. First transformation: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- [tex]\( B(-3,0) \rightarrow (-3, 0) \)[/tex]
- [tex]\( C(2,-1) \rightarrow (2, 1) \)[/tex]
- [tex]\( D(-1,2) \rightarrow (-1, -2) \)[/tex]
2. Second transformation: [tex]\((x, -y) \rightarrow (x+2, y+2)\)[/tex]
- [tex]\( B(-3,0) \rightarrow (-3+2, 0+2) = (-1, 2) \)[/tex]
- [tex]\( C(2,1) \rightarrow (2+2, 1+2) = (4, 3) \)[/tex]
- [tex]\( D(-1,-2) \rightarrow (-1+2, -2+2) = (1, 0) \)[/tex]
These vertices do not match our target vertices. Thus, Option 3 is not correct.
### Option 4: [tex]\((x, y) \rightarrow (-x, y) \rightarrow (x+2, y+2)\)[/tex]
1. First transformation: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- [tex]\( B(-3, 0) \rightarrow (3, 0) \)[/tex]
- [tex]\( C(2, -1) \rightarrow (-2, -1) \)[/tex]
- [tex]\( D(-1, 2) \rightarrow (1, 2) \)[/tex]
2. Second transformation: [tex]\(( - x, y) \rightarrow (x+2, y+2)\)[/tex]
- [tex]\( B(3, 0) \rightarrow (3+2, 0+2) = (5, 2) \)[/tex]
- [tex]\( C(-2, -1) \rightarrow (-2+2, -1+2) = (0, 1) \)[/tex]
- [tex]\( D(1, 2) \rightarrow (1+2, 2+2) = (3, 4) \)[/tex]
These vertices do not match our target vertices. Thus, Option 4 is not correct.
### Conclusion
Among all the given options, only Option 1 results in the desired transformed vertices. Therefore, the correct transformation is:
[tex]\[ \boxed{(x, y) \rightarrow (x+1, y+1) \rightarrow (y, x)} \][/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
