Answer :
To solve this problem, let's analyze the transformations applied to each vertex of triangle [tex]\( \triangle BCD \)[/tex] to determine how they match up with [tex]\( \triangle B' C' D' \)[/tex].
Given vertices:
- [tex]\( B(-3,0) \)[/tex]
- [tex]\( C(2,-1) \)[/tex]
- [tex]\( D(-1,2) \)[/tex]
Transformed vertices:
- [tex]\( B'(1,-2) \)[/tex]
- [tex]\( C'(0,3) \)[/tex]
- [tex]\( D'(3,0) \)[/tex]
We need to find which transformation matches the given results. Let's review each option step-by-step:
### 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]\( B(-3,0) \rightarrow (-2,1) \)[/tex]
- [tex]\( C(2,-1) \rightarrow (3,0) \)[/tex]
- [tex]\( D(-1,2) \rightarrow (0,3) \)[/tex]
2. Second transformation [tex]\((x, y) \rightarrow (y, x)\)[/tex]:
- [tex]\( (-2,1) \rightarrow (1,-2) \)[/tex]
- [tex]\( (3,0) \rightarrow (0,3) \)[/tex]
- [tex]\( (0,3) \rightarrow (3,0) \)[/tex]
After applying both transformations, we get:
- [tex]\( B'(-2, 1) \rightarrow B'(1, -2) \)[/tex]
- [tex]\( C'(3, 0) \rightarrow C'(0, 3) \)[/tex]
- [tex]\( D'(0, 3) \rightarrow D' (3, 0) \)[/tex]
This matches exactly with the transformed vertices [tex]\((B', C', D')\)[/tex].
### 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]\( B(-3,0) \rightarrow (-2,1) \)[/tex]
- [tex]\( C(2,-1) \rightarrow (3,0) \)[/tex]
- [tex]\( D(-1,2) \rightarrow (0,3) \)[/tex]
2. Second transformation [tex]\((x, y) \rightarrow (-x, y)\)[/tex]:
- [tex]\( (-2,1) \rightarrow (2,1) \)[/tex]
- [tex]\( (3,0) \rightarrow (-3,0) \)[/tex]
- [tex]\( (0,3) \rightarrow (0,3) \)[/tex]
This does not match the transformed vertices [tex]\((B', C', D')\)[/tex].
### 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]\( (-3,0) \rightarrow (-1,2) \)[/tex]
- [tex]\( (2,1) \rightarrow (4,3) \)[/tex]
- [tex]\( (-1,-2) \rightarrow (1,0) \)[/tex]
This does not match the transformed vertices [tex]\((B', C', D')\)[/tex].
### Option 4: [tex]\((x, y) \rightarrow (-x, y) \rightarrow (x+2, y+2+ \pi)\)[/tex]
This option includes a transformation involving [tex]\(\pi\)[/tex], which is uncommon and irrelevant in elementary transformations on the coordinate plane. This option doesn't match the nature of the given transformations.
Based on the above transformations, we conclude that the correct option is:
Option 1: [tex]\((x, y) \rightarrow (x+1, y+1) \rightarrow (y, x)\)[/tex].
Given vertices:
- [tex]\( B(-3,0) \)[/tex]
- [tex]\( C(2,-1) \)[/tex]
- [tex]\( D(-1,2) \)[/tex]
Transformed vertices:
- [tex]\( B'(1,-2) \)[/tex]
- [tex]\( C'(0,3) \)[/tex]
- [tex]\( D'(3,0) \)[/tex]
We need to find which transformation matches the given results. Let's review each option step-by-step:
### 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]\( B(-3,0) \rightarrow (-2,1) \)[/tex]
- [tex]\( C(2,-1) \rightarrow (3,0) \)[/tex]
- [tex]\( D(-1,2) \rightarrow (0,3) \)[/tex]
2. Second transformation [tex]\((x, y) \rightarrow (y, x)\)[/tex]:
- [tex]\( (-2,1) \rightarrow (1,-2) \)[/tex]
- [tex]\( (3,0) \rightarrow (0,3) \)[/tex]
- [tex]\( (0,3) \rightarrow (3,0) \)[/tex]
After applying both transformations, we get:
- [tex]\( B'(-2, 1) \rightarrow B'(1, -2) \)[/tex]
- [tex]\( C'(3, 0) \rightarrow C'(0, 3) \)[/tex]
- [tex]\( D'(0, 3) \rightarrow D' (3, 0) \)[/tex]
This matches exactly with the transformed vertices [tex]\((B', C', D')\)[/tex].
### 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]\( B(-3,0) \rightarrow (-2,1) \)[/tex]
- [tex]\( C(2,-1) \rightarrow (3,0) \)[/tex]
- [tex]\( D(-1,2) \rightarrow (0,3) \)[/tex]
2. Second transformation [tex]\((x, y) \rightarrow (-x, y)\)[/tex]:
- [tex]\( (-2,1) \rightarrow (2,1) \)[/tex]
- [tex]\( (3,0) \rightarrow (-3,0) \)[/tex]
- [tex]\( (0,3) \rightarrow (0,3) \)[/tex]
This does not match the transformed vertices [tex]\((B', C', D')\)[/tex].
### 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]\( (-3,0) \rightarrow (-1,2) \)[/tex]
- [tex]\( (2,1) \rightarrow (4,3) \)[/tex]
- [tex]\( (-1,-2) \rightarrow (1,0) \)[/tex]
This does not match the transformed vertices [tex]\((B', C', D')\)[/tex].
### Option 4: [tex]\((x, y) \rightarrow (-x, y) \rightarrow (x+2, y+2+ \pi)\)[/tex]
This option includes a transformation involving [tex]\(\pi\)[/tex], which is uncommon and irrelevant in elementary transformations on the coordinate plane. This option doesn't match the nature of the given transformations.
Based on the above transformations, we conclude that the correct option is:
Option 1: [tex]\((x, y) \rightarrow (x+1, y+1) \rightarrow (y, x)\)[/tex].
