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) ?

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) ?
A. a reflection across the y-axis and a move to the left by 5 and down by 3
B. a reflection across the y-axis and a move to the right by 3 and down by 2
C. a reflection across the x-axis and a move to the left by 3 and up by 2
D. a rotation of 270 degrees clockwise and a move right by 3 and up by 2

GPT-4o mini · Answer

To determine the composition transformation rule that maps triangle \( \triangle LMN \) to triangle \( \triangle L"M"N" \), we need to evaluate the coordinates of the vertices of both triangles.

The vertices of triangle \( \triangle LMN \) are:

\( L(1,1) \)
\( M(7,2) \)
\( N(5,7) \)

The vertices of triangle \( \triangle L"M"N" \) are:

\( L"(2,-1) \)
\( M"(-4,0) \)
\( N"(-2,5) \)

Now let's analyze these transformations:

Find the transformation from \( L \) to \( L" \):
- \( L(1, 1) \to L"(2, -1) \).
- To transform \( L(1, 1) \) to \( L"(2, -1) \):
  - From \( x = 1 \) to \( x = 2 \): shift right by \( 1 \).
  - From \( y = 1 \) to \( y = -1 \): shift down by \( 2 \).
Find the transformation from \( M \) to \( M" \):
- \( M(7, 2) \to M"(-4, 0) \).
- To transform \( M(7, 2) \) to \( M"(-4, 0) \):
  - From \( x = 7 \) to \( x = -4 \): shift left by \( 11 \).
  - From \( y = 2 \) to \( y = 0 \): shift down by \( 2 \).
Find the transformation from \( N \) to \( N" \):
- \( N(5, 7) \to N"(-2, 5) \).
- To transform \( N(5, 7) \) to \( N"(-2, 5) \):
  - From \( x = 5 \) to \( x = -2 \): shift left by \( 7 \).
  - From \( y = 7 \) to \( y = 5 \): shift down by \( 2 \).

Now, these transformations are not uniform, which suggests that there may be a reflection involved before the translation.

Considering the options given, we can check reflections:

Testing the reflection across the y-axis:
- Reflecting the points \( L, M, N \) across the y-axis:
  - \( L(1, 1) \) becomes \( L'(-1, 1) \).
  - \( M(7, 2) \) becomes \( M'(-7, 2) \).
  - \( N(5, 7) \) becomes \( N'(-5, 7) \).

However, the resulting coordinates are still distant from the final coordinates, as we concluded transformations requires translating or shifting the points afterward.

From the transformations observed, it seems that a combination of the reflection across the x-axis leads to choosing option C primarily since two point transformations lead to effective shift alignment.

Thus, considering these transformations holistically:

The transformation that maps \( \triangle LMN \) to \( \triangle L"M"N" \) is:

A reflection across the x-axis, followed by moving left and down adjustments.

This corresponds to choice: C. a reflection across the x-axis and a move to the left by 3 and up by 2. ⟶ Note that actual transformations may differ in dimension count but refers to effective movement. Hence check remaining dimensions from start to end.