To find the scale factor of the dilation, we can analyze the change in vertices from the original quadrilateral to the new quadrilateral. We'll do this by comparing a point from the original quadrilateral with its corresponding point on the new quadrilateral.
Let's take the first point for this purpose:
- Original Point: \( (4, 4) \)
- New Point: \( (3, 3) \)
The scale factor \( k \) can be determined by comparing the coordinates of the points. The scale factor can be calculated using the following formula: \[ k = \frac{\text{New coordinate}}{\text{Original coordinate}} \]
Calculating the scale factor for the x-coordinate: \[ k_x = \frac{3}{4} \]
Calculating the scale factor for the y-coordinate: \[ k_y = \frac{3}{4} \]
Since the scale factors for both coordinates are the same, we can conclude that the scale factor is consistent across this point.
We can double-check by using another pair of corresponding points. Let's take the second original point \( (6, 6) \) and its corresponding new point \( (4.5, 4.5) \):
Calculating:
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For the x-coordinate: \[ k_x = \frac{4.5}{6} = \frac{3}{4} \]
-
For the y-coordinate: \[ k_y = \frac{4.5}{6} = \frac{3}{4} \]
Both calculations yield the same scale factor \( k = \frac{3}{4} \).
We can proceed to check one more pair of points for further validation, such as the third original point \( (8, 4) \) and its corresponding new point \( (6, 3) \):
Calculating:
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For the x-coordinate: \[ k_x = \frac{6}{8} = \frac{3}{4} \]
-
For the y-coordinate: \[ k_y = \frac{3}{4} = \frac{3}{4} \]
Again, we obtain the same scale factor \( k = \frac{3}{4} \).
Finally, we can check the last pair, \( (6, 2) \) and \( (4.5, 1.5) \):
Calculating:
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For the x-coordinate: \[ k_x = \frac{4.5}{6} = \frac{3}{4} \]
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For the y-coordinate: \[ k_y = \frac{1.5}{2} = \frac{3}{4} \]
All pairs confirm that the scale factor used for the dilation of the quadrilateral is: \[ \boxed{\frac{3}{4}} \]