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

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 $\mathrm{△}LMN$ with vertices $L(1,1)$, $M(7,2)$, and $N(5,7)$ onto triangle $\mathrm{△}L"M"N"$ 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 $L^{\prime }(-1,1)$
2. $M(7,2)$ becomes $M^{\prime }(-7,2)$
3. $N(5,7)$ becomes $N^{\prime }(-5,7)$

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

1. $L^{\prime }(-1,1)$ becomes $L"(2,-1)$
2. $M^{\prime }(-7,2)$ becomes $M"(-4,0)$
3. $N^{\prime }(-5,7)$ 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.