Given the following points determine whether the transformation was a Rotation(specify direction and size), a reflection(over what point/line) or trasnlation(specify the rule).
1.
C(0,4). C'(2,6) N(-2,3) N'(0,5)
2.
G(0,0) G'(4,0) R(-1,4) R'(5,4)
3.
K(-4,-2) K'(4,-2) V(1,1) V'(9,1)
To determine whether the given transformations are rotations, reflections, or translations, we need to examine the changes in the coordinates of the points.
1. In this case, we have the following corresponding points:
C(0,4) -> C'(2,6)
N(-2,3) -> N'(0,5)
To determine the transformation, we can analyze the changes in the x-coordinates and y-coordinates separately.
For the x-coordinates:
C to C': Δx = 2 - 0 = 2
N to N': Δx = 0 - (-2) = 2
For the y-coordinates:
C to C': Δy = 6 - 4 = 2
N to N': Δy = 5 - 3 = 2
The Δx values and Δy values are the same, indicating a translation. Since the x-coordinates and y-coordinates increase by the same amount, we can determine the rule of the translation. In this case, the rule is: (x, y) -> (x + 2, y + 2). So, it is a translation by 2 units to the right and 2 units upwards.
2. For the second case, we have the following corresponding points:
G(0,0) -> G'(4,0)
R(-1,4) -> R'(5,4)
Analyzing the changes in the x-coordinates and y-coordinates separately:
For the x-coordinates:
G to G': Δx = 4 - 0 = 4
R to R': Δx = 5 - (-1) = 6
For the y-coordinates:
G to G': Δy = 0 - 0 = 0
R to R': Δy = 4 - 4 = 0
Since the Δx values are not equal for both corresponding points and the Δy values are equal, it indicates a reflection. The line of reflection can be determined by finding the mid-point of the two points. In this case, the mid-point is (2,4), so the line of reflection is the line x = 2.
3. Lastly, for the third case, we have the following corresponding points:
K(-4,-2) -> K'(4,-2)
V(1,1) -> V'(9,1)
Examining the changes in the x-coordinates and y-coordinates separately:
For the x-coordinates:
K to K': Δx = 4 - (-4) = 8
V to V': Δx = 9 - 1 = 8
For the y-coordinates:
K to K': Δy = -2 - (-2) = 0
V to V': Δy = 1 - 1 = 0
Since the Δx values are equal and the Δy values are equal, it indicates a rotation. To determine the direction and size of rotation, we can compare the changes in the x-coordinates and y-coordinates. In this case, both the x-coordinates and y-coordinates have increased by 8 units. Therefore, it is a rotation of 8 units counter-clockwise.