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.
To find the translation rule, we can calculate the difference in the x-values and the difference in the y-values between the corresponding points (L and L", M and M", N and N").
For L to L":
- The x-value of L" is 2, which is 1 unit greater than the x-value of L.
- The y-value of L" is -1, which is 2 units less than the y-value of L.
So, the translation rule for the x-values is +1 and for the y-values is -2.
For M to M":
- The x-value of M" is -4, which is 11 units less than the x-value of M.
- The y-value of M" is 0, which is the same as the y-value of M.
So, the translation rule for the x-values is -11 and for the y-values is 0.
For N to N":
- The x-value of N" is -2, which is 7 units less than the x-value of N.
- The y-value of N" is 5, which is the same as the y-value of N.
So, the translation rule for the x-values is -7 and for the y-values is 0.
Therefore, the composition transformation rule is:
△LMN → △L"M"N":
- x -> x + 1
- y -> y - 2
To find the composition transformation rule from triangle △LMN to triangle △L"M"N", we can use the formula:
(x', y') = T(x, y)
where T is the transformation rule, (x, y) are the coordinates of a point in the original triangle, and (x', y') are the coordinates of the corresponding point in the transformed triangle.
Let's find the transformation rule for each point:
1. For point L:
(x', y') = T(1, 1)
To find x', we subtract the x-coordinate of the original point L from the x-coordinate of the corresponding point L':
x' = 2 - 1 = 1
To find y', we subtract the y-coordinate of the original point L from the y-coordinate of the corresponding point L':
y' = -1 - 1 = -2
So, the coordinates of L" are (1, -2).
2. For point M:
(x', y') = T(7, 2)
To find x', we subtract the x-coordinate of the original point M from the x-coordinate of the corresponding point M':
x' = -4 - 7 = -11
To find y', we subtract the y-coordinate of the original point M from the y-coordinate of the corresponding point M':
y' = 0 - 2 = -2
So, the coordinates of M" are (-11, -2).
3. For point N:
(x', y') = T(5, 7)
To find x', we subtract the x-coordinate of the original point N from the x-coordinate of the corresponding point N':
x' = -2 - 5 = -7
To find y', we subtract the y-coordinate of the original point N from the y-coordinate of the corresponding point N':
y' = 5 - 7 = -2
So, the coordinates of N" are (-7, -2).
Therefore, the composition transformation rule that maps triangle △LMN onto triangle △L"M"N" is:
T(x, y) = (x-1, y-2)
To determine the composition transformation rule that maps △LMN onto △L"M"N", we need to find the translation, rotation, and/or reflection that takes the points L, M, and N to their corresponding points L", M", and N".
Let's start by analyzing the translation component of the transformation.
Translation: A translation shifts an object horizontally and/or vertically without changing its size, shape, or orientation. To find the translation vector, we subtract the corresponding coordinates of L and L", M and M", and N and N".
To find the translation vector for the x-coordinate, we subtract the x-coordinate of L" from L:
Δx = 2 - 1 = 1
To find the translation vector for the y-coordinate, we subtract the y-coordinate of L" from L:
Δy = -1 - 1 = -2
Since both Δx and Δy are positive (1 and -2), the translation vector is (1, -2).
Next, let's check if there is any rotation involved in the transformation.
Rotation: A rotation turns an object around a fixed point called the center of rotation. If there is no rotation, the center of rotation will be the same as the center of the original triangle.
Now, let's analyze the reflection component of the transformation.
Reflection: A reflection flips an object across an axis, usually a line. To check if there is a reflection, we need to compare the slopes of the line connecting L and M with the line connecting L" and M".
The slope of the line connecting L and M:
m1 = (2 - 1) / (7 - 1) = 1/6
The slope of the line connecting L" and M":
m2 = (0 - (-1)) / ((-4) - 2) = 1/6
Since the slopes are the same, there is no reflection component in the transformation.
Based on our analysis, the transformation from △LMN to △L"M"N" consists of a translation with a vector (1, -2). Thus, the composition transformation rule is:
(x, y) → (x + 1, y - 2)
Therefore, to map any point (x, y) in △LMN to △L"M"N", you would add 1 to the x-coordinate and subtract 2 from the y-coordinate.