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)
?
The composition transformation rule that maps △LMN onto △L"M"N" is a translation.
options are
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
a reflection across the y -axis and a move to the right by 3 and down by 2
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 y-axis and a move to the left by 5 and up by 3
The correct option is "a reflection across the y-axis and a move to the left by 3 and down by 2".
To determine the composition transformation rule that maps △LMN to △L"M"N", we need to find the translation, rotation, and reflection involved.
Translation:
The translation involved can be found by subtracting the corresponding coordinates of L and L", M and M", and N and N":
Δx = L"x - Lx = 2 - 1 = 1
Δy = L"y - Ly = -1 - 1 = -2
Therefore, the translation is (1, -2).
Rotation:
To determine the rotation, we can compare the slopes of LM and L"M".
Slope of LM:
m(LM) = (My - Ly) / (Mx - Lx)
= (2 - 1) / (7 - 1)
= 1/6
Slope of L"M":
m(L"M") = (M"y - L"y) / (M"x - L"x)
= (-1 - (-2)) / (2 - (-4))
= 1/6
Since both slopes are equal, we can conclude that there is no rotation involved.
Reflection:
To determine the reflection, we can compare the coordinates of M and M".
Comparing the x-coordinates:
M"x = Mx = 7
Comparing the y-coordinates:
M"y = Ly - (My - Ly) = 1 - (2 - 1) = 0
Since both the x and y-coordinates are the same, we can conclude that there is no reflection involved.
Therefore, the composition transformation rule that maps △LMN to △L"M"N" is a translation of (1, -2).
To determine the composition transformation rule that maps △LMN to △L"M"N", we need to find the sequence of transformations applied to the original triangle.
Step 1: Translation
First, let's analyze the translation. We can find the translation vector by subtracting the corresponding coordinates of L and L":
Translation vector = (x_L" - x_L, y_L" - y_L) = (2 - 1, -1 - 1) = (1, -2)
So, the translation is (1, -2).
Step 2: Reflection
Next, we need to determine if there is a reflection involved. To do this, we can compare the slopes of the lines connecting the corresponding vertices of the original and final triangles.
The slope of LM (original) = (y_M - y_L) / (x_M - x_L) = (2 - 1) / (7 - 1) = 1 / 6
The slope of L"M" (final) = (y_M" - y_L") / (x_M" - x_L") = (0 - (-1)) / ((-4) - 2) = 1 / 6
The slopes are equal, which means there is no reflection.
Step 3: Scaling
To determine if there is scaling, we can compare the lengths of the corresponding sides of the original and final triangles.
Length of LM (original) = √((x_M - x_L)^2 + (y_M - y_L)^2) = √((7 - 1)^2 + (2 - 1)^2) = √(36 + 1) = √37
Length of L"M" (final) = √((x_M" - x_L")^2 + (y_M" - y_L")^2) = √((-4 - 2)^2 + (0 - (-1))^2) = √((-6)^2 + 1) = √37
The lengths are equal, which means there is no scaling.
Step 4: Rotation
Lastly, we need to determine if there is a rotation. To do this, we can calculate the angles between the corresponding sides of the original and final triangles.
Angle between ML and M"L" = arctan((y_L" - y_M") / (x_L" - x_M")) = arctan((-1 - 0) / (2 - (-4))) = arctan(-1/6) = -9.46 degrees
The angle is not equal to 0 or 180 degrees, which means there is a rotation.
Summary:
Based on our analysis, the composition transformation rule for △LMN to △L"M"N" is as follows:
1. Translation: (1, -2)
2. Rotation: -9.46 degrees (counter-clockwise)
Please note that the angles and direction of rotation might differ based on the conventions used in your specific context.