What is the scale factor of this dilation if the points have went from (-1,1), (-2,-2), (2,-2), and (3,1) to (-3,3), (-6,-6), (6,-6), and (9,3)?
To find the scale factor of the dilation, we can choose any pair of corresponding points and find the ratio of their distances. Let's choose (-1,1) and (-3,3):
The distance between (-1,1) and (-3,3) in the original set of points is sqrt((-3-(-1))^2 + (3-1)^2) = sqrt((-2)^2 + (2)^2) = sqrt(4 + 4) = sqrt(8).
The distance between (-1,1) and (-3,3) in the dilated set of points is sqrt((-3-(-1))^2 + (3-1)^2) = sqrt((-2)^2 + (2)^2) = sqrt(4 + 4) = sqrt(8).
The ratio of the distances is sqrt(8)/sqrt(8) = 1.
Therefore, the scale factor of this dilation is 1.
To find the scale factor of a dilation, we divide the change in one coordinate (either x or y) of a point in the image by the corresponding change in the same coordinate of the corresponding point in the pre-image.
Let's calculate the change in the x-coordinate of the first point:
Change in x-coordinate = x-coordinate of the image point - x-coordinate of the pre-image point
= (-3) - (-1)
= -3 + 1
= -2
Now, let's calculate the change in the x-coordinate of the corresponding point in the pre-image:
Change in x-coordinate = x-coordinate of the corresponding pre-image point - x-coordinate of the previous pre-image point
= (-2) - (-1)
= -2 + 1
= -1
The scale factor for the x-coordinate is the ratio of these two changes:
Scale factor (x-coordinate) = Change in x-coordinate of the image / Change in x-coordinate of the corresponding pre-image
= -2 / -1
= 2
Similarly, we can find the scale factors for the y-coordinate by following the same steps:
Change in y-coordinate of the first point = y-coordinate of the image point - y-coordinate of the pre-image point
= 3 - 1
= 2
Change in y-coordinate of the corresponding point in the pre-image = (-6) - (-2)
= -6 + 2
= -4
Scale factor (y-coordinate) = Change in y-coordinate of the image / Change in y-coordinate of the corresponding pre-image
= 2 / -4
= -0.5
Therefore, the scale factor of this dilation is 2 for the x-coordinate and -0.5 for the y-coordinate.
To find the scale factor of the dilation, we need to compare the corresponding distances between the points before and after the dilation.
Let's take the first pair of points: (-1,1) and (-3,3). The x-coordinate has gone from -1 to -3, which is a decrease of 2 units. Similarly, the y-coordinate has gone from 1 to 3, a increase of 2 units. Since both the x and y coordinates have changed by 2 units in opposite directions, we can conclude that the scale factor is 2. This means that the image is double the size of the original object.
We can repeat this process for the other pairs of points to confirm that the scale factor is consistent.
For the second pair: (-2,-2) and (-6,-6). The x-coordinate has gone from -2 to -6, which is again a decrease of 4 units. The y-coordinate has also decreased by 4 units. Therefore, the scale factor is 4.
For the third pair: (2,-2) and (6,-6). The x-coordinate has gone from 2 to 6, an increase of 4 units, and the y-coordinate has decreased by 4 units. So, the scale factor is 4.
Finally, for the fourth pair: (3,1) and (9,3). The x-coordinate has increased by 6 units, and the y-coordinate has increased by 2 units. The scale factor is still 3.
In conclusion, the scale factor of this dilation is 2.