Answered

A triangle has vertices at [tex]\( B(-3,0), C(2,-1), D(-1,2) \)[/tex]. Which transformation would produce an image with vertices [tex]\( B^{\circ}(1,-2), C^{-}(0,3), D^*(3,0) \)[/tex]?

A. [tex]\((x, y) \rightarrow (x+1, y+1) \rightarrow (y, x)\)[/tex]
B. [tex]\((x, y) \rightarrow (x+1, y+1) \rightarrow (-x, y)\)[/tex]
C. [tex]\((x, y) \rightarrow (x, -y) \rightarrow (x+2, y+2)\)[/tex]
D. [tex]\((x, y) \rightarrow (-x, y) \rightarrow (x+2, y+2)\)[/tex]



Answer :

GinnyAnswer
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]

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