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 transformation was a rotation, reflection, or translation, we need to analyze the given points and their corresponding points after the transformation.
1.
C(0, 4), C'(2, 6), N(-2, 3), N'(0, 5)
To determine if it was a rotation, we need to check if the distances between the points and their corresponding points remain the same. Calculate the distances between the points for both sets:
Distance(C, N) = sqrt((0 - (-2))^2 + (4 - 3)^2) = sqrt(4 + 1) = sqrt(5)
Distance(C', N') = sqrt((2 - 0)^2 + (6 - 5)^2) = sqrt(4 + 1) = sqrt(5)
Since the distances between the points remain the same, it indicates that there might be a rotation. To determine the direction and size of the rotation, we need to find the center of rotation.
Center of rotation = midpoint(C, C') = ((0 + 2)/2, (4 + 6)/2) = (1, 5)
Based on the center of rotation, we can see that the points have rotated clockwise from the original positions. Therefore, the transformation is a rotation clockwise about the point (1, 5) with a size of sqrt(5).
2.
G(0, 0), G'(4, 0), R(-1, 4), R'(5, 4)
To determine if it was a reflection, we need to check if the y-coordinates of the points and their corresponding points are the same, while the x-coordinates differ by equal amounts.
Comparing the y-coordinates:
G(0, 0), G'(4, 0) --> The y-coordinate remains the same (0).
R(-1, 4), R'(5, 4) --> The y-coordinate remains the same (4).
Comparing the x-coordinates:
G(0, 0), G'(4, 0) --> The x-coordinates differ by 4.
R(-1, 4), R'(5, 4) --> The x-coordinates differ by 6.
Since the y-coordinates remain the same and the x-coordinates differ by equal amounts, it indicates that there might be a reflection. To determine the line of reflection, we can consider the midpoint between the original points and the corresponding points.
Line of reflection = midpoint(G, G') = ((0 + 4)/2, (0 + 0)/2) = (2, 0)
Based on the line of reflection, we can see that the points have reflected across the x-axis. Therefore, the transformation is a reflection over the x-axis.
3.
K(-4, -2), K'(4, -2), V(1, 1), V'(9, 1)
To determine if it was a translation, we need to check if the distances and directions between the points and their corresponding points are the same.
Distance(K, K') = sqrt((4 - (-4))^2 + (-2 - (-2))^2) = sqrt(64) = 8
Distance(V, V') = sqrt((9 - 1)^2 + (1 - 1)^2) = sqrt(64) = 8
Since the distances between the points remain the same and the directions are consistent, it indicates that there might be a translation. To determine the rule of translation, we can consider the difference between the coordinates of the original points and the corresponding points.
Translation rule = (x-coordinate of K' - x-coordinate of K, y-coordinate of K' - y-coordinate of K)
= (4 - (-4), -2 - (-2))
= (8, 0)
Based on the translation rule, we can see that the points have shifted 8 units to the right. Therefore, the transformation is a translation 8 units to the right.