What composition transformation rule has △LMN

Question

△
𝐿
𝑀
𝑁
, L(1,1)
𝐿
(
1
,
1
)
, M(7,2)
𝑀
(
7
,
2
)
, and N(5,7)
𝑁
(
5
,
7
)
map onto △L"M"N"
△
𝐿
"
𝑀
"
𝑁
"
, L"(2,−1)
𝐿
"
(
2
,
−
1
)
, M"(−4,0)
𝑀
"
(
−
4
,
0
)
, and N"(−2,5)
𝑁
"
(
−
2
,
5
)
? (1 point)
Responses

a reflection across the y-axis and a move to the left by 5 and up by 3
a reflection across the y -axis and a move to the left by 5 and up by 3

a rotation of 270 degrees clockwise and a move right by 3 and up by 2
a rotation of 270 degrees clockwise and a move right by 3 and up by 2

a reflection across the x-axis and a move to the left by 3 and down by 2
a reflection across the x -axis and a move to the left by 3 and down by 2

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

Answer 1

To determine the appropriate transformation 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 will analyze the changes in the coordinates.

1. Original vertices:

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

2. Transformed vertices:

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

Now, let's determine how each vertex changes.

Change for L:

• From \( (1,1) \) to \( (2,-1) \):
  • \( x \): \( 1 \to 2 \) (increase by 1)
  • \( y \): \( 1 \to -1 \) (decrease by 2)

Change for M:

• From \( (7,2) \) to \( (-4,0) \):
  • \( x \): \( 7 \to -4 \) (decrease by 11)
  • \( y \): \( 2 \to 0 \) (decrease by 2)

Change for N:

• From \( (5,7) \) to \( (-2,5) \):
  • \( x \): \( 5 \to -2 \) (decrease by 7)
  • \( y \): \( 7 \to 5 \) (decrease by 2)

By analyzing the changes, we see that there appears to be a reflection and a standard transformation in play. The x-coordinates show that all undergo shifts in opposite directions, which hints at a reflection.

Summary of transformations: To get from \( \triangle LMN \) to \( \triangle L"M"N" \):

1. Reflection across the y-axis: This will change the signs of the x-coordinates.
2. Vertical translation: The points altered their y-coordinates.

Let’s revise the final transformation—checking both:

• The y-axis reflection changes \( (x,y) \) to \( (-x,y) \).
• Adjusting the result by moving to the right by 3 and down by 2 will account for the changes necessary to match \( L", M", N" \).

Thus, the best fit from your options for the transformation from \( \triangle LMN \) to \( \triangle L"M"N" \) is:

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