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

Answer 1

To find the composition transformation rule that maps triangle \( \triangle LMN \) with vertices \( L(1,1) \), \( M(7,2) \), and \( N(5,7) \) onto triangle \( \triangle L"M"N" \) with vertices \( L"(2,-1) \), \( M"(-4,0) \), and \( N"(-2,5) \), we can analyze the transformations step by step.

Let's first look at the coordinates:

- \( L(1, 1) \) to \( L"(2, -1) \)
- \( M(7, 2) \) to \( M"(-4, 0) \)
- \( N(5, 7) \) to \( N"(-2, 5) \)

### Step 1: Determine if there is a reflection

Let's determine if the transformation involves a reflection by checking how the points change.

- For point \( L \): from \( (1,1) \) to \( (2,-1) \):
- The change in the x-coordinate is \( 2 - 1 = 1 \) (moves right by 1).
- The change in the y-coordinate is \( -1 - 1 = -2 \) (moves down by 2).

- For point \( M \): from \( (7,2) \) to \( (-4,0) \):
- The change in the x-coordinate is \( -4 - 7 = -11 \) (moves left by 11).
- The change in the y-coordinate is \( 0 - 2 = -2 \) (moves down by 2).

- For point \( N \): from \( (5,7) \) to \( (-2,5) \):
- The change in the x-coordinate is \( -2 - 5 = -7 \) (moves left by 7).
- The change in the y-coordinate is \( 5 - 7 = -2 \) (moves down by 2).

### Step 2: Determine the nature of the reflection

Considering that the points exhibit a significant change in the x-coordinates for \( M \), particularly, we can see that reflecting across the y-axis (which would negate the x-coordinates) could also explain the large movements.

Reflecting across the y-axis:

1. \( L(1, 1) \) becomes \( L'(-1, 1) \)
2. \( M(7, 2) \) becomes \( M'(-7, 2) \)
3. \( N(5, 7) \) becomes \( N'(-5, 7) \)

Then adding a translation of \( (+3, -2) \):

1. \( L'(-1, 1) \) becomes \( L"(2, -1) \)
2. \( M'(-7, 2) \) becomes \( M"(-4, 0) \)
3. \( N'(-5, 7) \) becomes \( N"(-2, 5) \)

This matches perfectly with the coordinates of the transformed triangle.

### Conclusion

The transformations confirm that:

- **Reflection across the y-axis**
- **Translation 3 units to the right and 2 units down**

Thus, the correct answer to the composition transformation rule is:

**A. a reflection across the y-axis and a move to the right by 3 and down by 2.**