What is the scale factor of the dilation where the points have gone from (-0.5,2), (-0.5,-1), (1.5,2), and (1.5,-1) to (-5,9), (3,9), (-5,-4), and (3,-4)?
A. 2
B. 1/2
C. 4
D. 1/4
To find the scale factor of a dilation, we can compare the distances between corresponding points in the pre-image and the image.
The distance between (-0.5, 2) and (-0.5, -1) is 2 - (-1) = 3 units.
The distance between (-5, 9) and (3, 9) is 3 - (-5) = 8 units.
Therefore, the scale factor for the x-direction is 8/3.
The distance between (-0.5, 2) and (1.5, 2) is 1.5 - (-0.5) = 2 units.
The distance between (-5, 9) and (-5, -4) is 9 - (-4) = 13 units.
Therefore, the scale factor for the y-direction is 13/2.
Since the scale factor for the x-direction is 8/3 and the scale factor for the y-direction is 13/2, the overall scale factor is (8/3) * (13/2) = 104/6 = 52/3.
Therefore, the scale factor of the dilation is $\boxed{\text{(C) }4}$.
To find the scale factor of the dilation, we can compare the distance between corresponding points before and after the dilation.
Let's take two corresponding points from the original and dilated sets of points:
1. (-0.5, 2) and (-5, 9)
To find the distance between these two points, we can use the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((-5 - (-0.5))^2 + (9 - 2)^2)
= sqrt((-4.5)^2 + 7^2)
= sqrt(20.25 + 49)
= sqrt(69.25)
≈ 8.32
Now, let's take the corresponding points (1.5, 2) and (3, 9):
Distance = sqrt((3 - 1.5)^2 + (9 - 2)^2)
= sqrt((1.5)^2 + 7^2)
= sqrt(2.25 + 49)
= sqrt(51.25)
≈ 7.15
To find the scale factor, we divide the distance between the corresponding points after the dilation by the distance between the corresponding points before the dilation:
Scale Factor = Distance After Dilation / Distance Before Dilation
Scale Factor = 7.15 / 8.32
≈ 0.86
Rounded to the nearest hundredth, the scale factor is approximately 0.86.
None of the given options (A, B, C, D) matches this value, so none of the given options is the correct scale factor for this dilation.
To find the scale factor of a dilation, we can compare the distances between corresponding points before and after the dilation.
Let's consider the point (-0.5, 2) before the dilation and (-5, 9) after the dilation. The x-coordinate of the point (-5, 9) is 10 times smaller than the x-coordinate of (-0.5, 2), and the y-coordinate of the point (-5, 9) is 4.5 times greater than the y-coordinate of (-0.5, 2). Therefore, we can conclude that the dilation has a horizontal scale factor of 1/10 and a vertical scale factor of 4.5.
Now, let's consider the point (1.5, 2) before the dilation and (3, -4) after the dilation. The x-coordinate of the point (3, -4) is 2 times smaller than the x-coordinate of (1.5, 2), and the y-coordinate of the point (3, -4) is 2 times smaller than the y-coordinate of (1.5, 2). Therefore, we can conclude that the dilation has a horizontal scale factor of 1/2 and a vertical scale factor of 1/2.
Since the scale factors for both the x and y directions are the same for all points, we can say that the scale factor of the dilation is the common scale factor, which is 1/2.
Thus, the answer is B. 1/2.