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)?
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.
To find the scale factor of a dilation, we can use the formula:
scale factor = (distance between corresponding points after dilation) / (distance between corresponding points before dilation)
Let's calculate the scale factor step-by-step:
1. Calculate the distance between the corresponding points before and after dilation.
Before dilation:
- Distance between (-1, 2) and (-1, -1) = |-1 - (-1)| = 0
- Distance between (2, 2) and (2, -1) = |-1 - 2| = 3
After dilation:
- Distance between (-5, 9) and (3, 9) = |3 - (-5)| = 8
- Distance between (-5, -4) and (3, -4) = |-4 - (-4)| = 0
2. Divide the distances after dilation by the distances before dilation to find the scale factor.
- Scale factor = (8 / 0) / (0 / 3) = undefined (since we cannot divide by zero)
The scale factor is undefined in this case since one of the distances before dilation is zero.
To find the scale factor of a dilation, we need to compare the corresponding distances between the original points and the dilated points.
Let's take the first point (-1, 2) and its corresponding dilated point (-5, 9).
The horizontal distance between the two points is |-1 - (-5)| = 4.
The vertical distance between the two points is |2 - 9| = 7.
Now let's take the second point (-1, -1) and its corresponding dilated point (3, 9).
The horizontal distance between the two points is |-1 - 3| = 4.
The vertical distance between the two points is |-1 - 9| = 10.
Continuing with the remaining two pairs of points, we find:
The horizontal distances between the third points (2, 2) and (2, -1) and their corresponding dilated points (-5, -4) and (3, -4) are both 7.
The vertical distances between the third points (2, 2) and (2, -1) and their corresponding dilated points (-5, -4) and (3, -4) are both |-4 - 2| = 6.
Now, we need to find the ratio of the corresponding distances. Let's take the horizontal distance:
For the first point, the horizontal distance ratio is 4/4 = 1.
For the second point, the horizontal distance ratio is 4/4 = 1.
For the third and fourth points, the horizontal distance ratio is 7/7 = 1.
Since the horizontal distance ratio is the same for all point pairs, we can conclude that the scale factor for the horizontal direction is 1.
Now, let's find the vertical distance ratio:
For the first point, the vertical distance ratio is 7/7 = 1.
For the second point, the vertical distance ratio is 10/7 ≈ 1.43.
For the third and fourth points, the vertical distance ratio is 6/7 ≈ 0.86.
Since the vertical distance ratio is not the same for all point pairs, we cannot determine a single scale factor for the vertical direction. The scale factors for the vertical direction are different for each pair of points.
Therefore, the scale factor for the dilation in this case is 1 for the horizontal direction, and the vertical scale factors vary depending on the point pair.