What composition transformation rule has applied to have △LMN map onto ​△L′M′N′ , if the coordinates are: ​​​​ L(8,3) , M(4,6) , and N(5,9) L′(−10,−3) , M′(−6,0) , and N′(−7,3) ? (1 point) Responses a rotation of 270 degrees clockwise and a move right by 2 and up by 6 a rotation of 270 degrees clockwise and a move right by 2 and up by 6 a reflection over the y-axis and a move to the left by 2 and down by 6 a reflection over the y-axis and a move to the left by 2 and down by 6 a reflection over the y-axis and a move to the left by 6 and down by 2 a reflection over the y-axis and a move to the left by 6 and down by 2 a reflection across the x-axis and a move to the right by 2 and down by 6

Answer 1

To determine the transformation that maps triangle \( \triangle LMN \) onto triangle \( \triangle L'M'N' \), we first need to examine the coordinates of the points before and after the transformation.

The coordinates are as follows:

• \( L(8, 3) \) transforms to \( L'(-10, -3) \)
• \( M(4, 6) \) transforms to \( M'(-6, 0) \)
• \( N(5, 9) \) transforms to \( N'(-7, 3) \)

Step 1: Check for Reflection

Let's check if there is a reflection involved by considering the y-axis reflection, which negates the x-coordinates of the points.

1. If we reflect over the y-axis:
   • \( L(8, 3) \) becomes \( L'(-8, 3) \)
   • \( M(4, 6) \) becomes \( M'(-4, 6) \)
   • \( N(5, 9) \) becomes \( N'(-5, 9) \)

This shows that reflecting over the y-axis does not yield the corresponding points we have.

Step 2: Check for Reflection over the x-axis

Next, we will check for a reflection over the x-axis:

1. If we reflect over the x-axis:
   • \( L(8, 3) \) becomes \( L'(8, -3) \)
   • \( M(4, 6) \) becomes \( M'(4, -6) \)
   • \( N(5, 9) \) becomes \( N'(5, -9) \)

This again does not yield the corresponding points.

Step 3: Check for Movement

Since reflections do not work, let's analyze the translation of the points after reflecting them. If points are reflected and then translated, the movement translations should still hold.

After each point has notably changed in both coordinates, let's check:

• Changes in L:

  • From \( (8, 3) \) to \( (-10, -3) \):
    • Change in x: \( -10 - 8 = -18 \)
    • Change in y: \( -3 - 3 = -6 \)

• Changes in M:

  • From \( (4, 6) \) to \( (-6, 0) \):
    • Change in x: \( -6 - 4 = -10 \)
    • Change in y: \( 0 - 6 = -6 \)

• Changes in N:

  • From \( (5, 9) \) to \( (-7, 3) \):
    • Change in x: \( -7 - 5 = -12 \)
    • Change in y: \( 3 - 9 = -6 \)

Summary

The major change is in x-coordinates being reflected and subsequently translated. Each original coordinate indeed shows a reflection over the y-axis, followed by various movements to get final positions as seen in the coordinates of \( L' M' N' \).

• There is a reflection over the y-axis and a uniform translation of moving to the left by 6 and down by 2.

Thus, the correct transformation is:

a reflection over the y-axis and a move to the left by 6 and down by 2.