Question

What is the scale factor of the dilation where the points have gone from (-1,2), (-1,-1), (2,2), and (2,-1) to (-5,9), (3,9), (-5,-4), and (3,-4)?

Answer 1

To find the scale factor of the dilation, we need to compare the corresponding distances between the original points and the dilated points.

The original points are (-1,2), (-1,-1), (2,2), and (2,-1).

Let's compare the x-coordinates of the points:
The x-coordinate of the first point has decreased by 4 units: -1 - (-5) = 4
The x-coordinate of the second point has increased by 4 units: -1 - 3 = -4
The x-coordinate of the third point has decreased by 7 units: 2 - (-5) = 7
The x-coordinate of the fourth point has increased by 1 unit: 2 - 3 = -1

Now, let's compare the y-coordinates of the points:
The y-coordinate of the first point has increased by 7 units: 2 - 9 = -7
The y-coordinate of the second point has increased by 10 units: -1 - 9 = -10
The y-coordinate of the third point has decreased by 6 units: 2 - (-4) = 6
The y-coordinate of the fourth point has decreased by 3 units: -1 - (-4) = 3

To find the scale factor, we can compare the change in x-coordinates to the change in y-coordinates for two corresponding points.

Comparing points (-1,2) and (-1,-1):
Change in x-coordinates: 4
Change in y-coordinates: -7

Comparing points (2,2) and (2,-1):
Change in x-coordinates: -7
Change in y-coordinates: 6

We can use either comparison to find the scale factor. Let's use the first comparison:
Change in x-coordinates: 4
Change in y-coordinates: -7

The scale factor is the absolute value of the ratio of the change in y-coordinates to the change in x-coordinates:
Scale factor = |change in y / change in x| = |-7 / 4| = 7/4 = 1.75

Therefore, the scale factor of the dilation is 1.75.