Question
What composition transformation rule has △LMN, L(1,1), M(7,2), and N(5,7) map onto △L"M"N", L"(2,−1), M"(−4,0), and N"(−2,5)? (1 point)Responsesa reflection across the y-axis and a move to the right by 3 and down by 2a reflection across the y -axis and a move to the right by 3 and down by 2a reflection across the y-axis and a move to the left by 5 and up by 3a reflection across the y -axis and a move to the left by 5 and up by 3a rotation of 270 degrees clockwise and a move right by 3 and up by 2a rotation of 270 degrees clockwise and a move right by 3 and up by 2a reflection across the x-axis and a move to the left by 3 and down by 2
Answers
GPT-4o mini
To determine which composition transformation rule maps triangle \( \triangle LMN \) with points \( L(1,1) \), \( M(7,2) \), and \( N(5,7) \) onto triangle \( \triangle L"M"N" \) with points \( L"(2,-1) \), \( M"(-4,0) \), and \( N"(-2,5) \), we need to analyze the transformations involved.
1. **Identify the transformations:**
Let's first find the movement of point \( L \):
- \( L(1,1) \) maps to \( L"(2,-1) \).
- This can be seen as a reflection across the y-axis followed by a translation.
Reflection across the y-axis changes the x-coordinate's sign:
- \( L'(x) = (-x, y) \)
- Thus, \( L'(1, 1) = (-1, 1) \).
From \( L'(-1, 1) \) to \( L"(2, -1) \):
- The translation vector from \( (-1, 1) \) to \( (2, -1) \) is:
\[
(2 - (-1), -1 - 1) = (3, -2).
\]
This means we move:
- Right by \( 3 \) (positive x-direction).
- Down by \( 2 \) (negative y-direction).
2. **Repeat for other points to confirm consistency:**
Now check points \( M \) and \( N \):
- \( M(7,2) \) should also obey the same rules.
- Reflect \( M(7, 2) \):
\[
M'(7, 2) = (-7, 2).
\]
- Translate \( M'(-7, 2) \) by \( (3, -2) \):
\[
M"(-7 + 3, 2 - 2) = (-4, 0).
\]
Now for point \( N \):
- Reflect \( N(5, 7) \):
\[
N'(5, 7) = (-5, 7).
\]
- Translate \( N'(-5, 7) \) by \( (3, -2) \):
\[
N"(-5 + 3, 7 - 2) = (-2, 5).
\]
All points \( L, M, N \) follow the same set of transformations thus confirming that the overall transformation is:
- A reflection across the y-axis followed by a translation of right \( 3 \) and down \( 2 \).
The correct answer is:
- A reflection across the y-axis and a move to the right by 3 and down by 2.
1. **Identify the transformations:**
Let's first find the movement of point \( L \):
- \( L(1,1) \) maps to \( L"(2,-1) \).
- This can be seen as a reflection across the y-axis followed by a translation.
Reflection across the y-axis changes the x-coordinate's sign:
- \( L'(x) = (-x, y) \)
- Thus, \( L'(1, 1) = (-1, 1) \).
From \( L'(-1, 1) \) to \( L"(2, -1) \):
- The translation vector from \( (-1, 1) \) to \( (2, -1) \) is:
\[
(2 - (-1), -1 - 1) = (3, -2).
\]
This means we move:
- Right by \( 3 \) (positive x-direction).
- Down by \( 2 \) (negative y-direction).
2. **Repeat for other points to confirm consistency:**
Now check points \( M \) and \( N \):
- \( M(7,2) \) should also obey the same rules.
- Reflect \( M(7, 2) \):
\[
M'(7, 2) = (-7, 2).
\]
- Translate \( M'(-7, 2) \) by \( (3, -2) \):
\[
M"(-7 + 3, 2 - 2) = (-4, 0).
\]
Now for point \( N \):
- Reflect \( N(5, 7) \):
\[
N'(5, 7) = (-5, 7).
\]
- Translate \( N'(-5, 7) \) by \( (3, -2) \):
\[
N"(-5 + 3, 7 - 2) = (-2, 5).
\]
All points \( L, M, N \) follow the same set of transformations thus confirming that the overall transformation is:
- A reflection across the y-axis followed by a translation of right \( 3 \) and down \( 2 \).
The correct answer is:
- A reflection across the y-axis and a move to the right by 3 and down by 2.