What composition transformation rule has applied to have △LMN△LMN map onto ​△L′M′N′​△L′M′N′, if the coordinates are: ​​​​ L(8,3)L(8,3), M(4,6)M(4,6), and N(5,9)N(5,9) L′(−10,−3)L′(−10,−3), M′(−6,0)M′(−6,0), and N′(−7,3)N′(−7,3)? (1 point)

Question

a reflection across the x-axis and a move to the right 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 2 and down by 6
a rotation of 270 degrees clockwise and a move right by 2 and up by 6

Answer 1

To determine the composition transformation that maps triangle \( \triangle LMN \) with vertices \( L(8,3) \), \( M(4,6) \), and \( N(5,9) \) onto triangle \( \triangle L'M'N' \) with vertices \( L'(-10,-3) \), \( M'(-6,0) \), and \( N'(-7,3) \), we need to first analyze the changes in the coordinates.

Step 1: Translate the vertices
Let's look at the transformation from \( \triangle LMN \) to \( \triangle L'M'N' \):

1. Coordinate of \( L \):
   \( L(8, 3) \) to \( L'(-10, -3) \)
   Change:

   • \( x \): \( -10 - 8 = -18 \)
   • \( y \): \( -3 - 3 = -6 \)

2. Coordinate of \( M \):
   \( M(4, 6) \) to \( M'(-6, 0) \)
   Change:

   • \( x \): \( -6 - 4 = -10 \)
   • \( y \): \( 0 - 6 = -6 \)

3. Coordinate of \( N \):
   \( N(5, 9) \) to \( N'(-7, 3) \)
   Change:

   • \( x \): \( -7 - 5 = -12 \)
   • \( y \): \( 3 - 9 = -6 \)

From this analysis, we can see that the change in the \( y \)-coordinates is consistent (all decrease by 6), while the change in the \( x \)-coordinates varies significantly.

Step 2: Identify the type of transformation
Since the change in the \( y \)-coordinates suggests a vertical reflection since it has flipped from positive to negative, we can check for reflection over the x-axis.

Reflection over the x-axis:

• Reversing the signs of the \( y \)-coordinates:
  • \( L(8, 3) \rightarrow L(8, -3) \)
  • \( M(4, 6) \rightarrow M(4, -6) \)
  • \( N(5, 9) \rightarrow N(5, -9) \)

After reflection across the x-axis, the points are \( (8,-3) \), \( (4,-6) \), \( (5,-9) \).

Step 3: Apply the translation
Now, we find the translation that takes:

• \( L(8, -3) \) to \( L'(-10,-3) \):
  Move left by \( -10 - 8 = -18 \) in \( x \) and the \( y \) remains (-3).
• After that, notice \( M(4, -6) \) to \( M'(-6, 0) \):
• Move left by \( -6 - 4 = -10 \) in the \( x \) shift which seems inconsistent with what was computed.

Thus, further analysis indicates that the transformation doesn't match any rigid transformation but suggests a close match with option (b).

The best interpretation, therefore, is:

• A reflection over the y-axis and then translation down results towards the desired positions is indeed the best match aligning pairs accurately down \( 2 \) consistently matches where they result finally.

Thus, the overall transformation will be:
Answer: a reflection over the y-axis and a move to the left by 2 and down by 6.