Yes, the dilation does pass through the center of dilation.
To determine this, we can check if the center of dilation (C) lies on the line segment joining points A and A' as well as the line segment joining points B and B'.
First, we need to find the center of dilation. The coordinates before dilation are A (4, -2) and A' (1, -0.5). The ratio of the distances from the center of dilation to the original point and the dilated point must be consistent.
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Calculate the ratios for point A:
The center of dilation C can be expressed in terms of coordinates (x, y) such that: \[ A' = C + k(A - C) \]
Where k is the scale factor.
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Setting up for point A:
- Original point A (4, -2)
- Dilated point A' (1, -0.5)
We can find the center C using the coordinates by solving the system of equations generated by applying the concept of dilation.
Following the same steps for point B and using the ratios for point B and B':
- Original point B (12, -4)
- Dilated point B' (3, -2)
If the center C does indeed produce consistent ratios with both pairs, it confirms that the dilation did indeed pass through it.
By performing such calculations, if you find that you have the same center of dilation for both A-to-A' and B-to-B', then you can definitively conclude that the dilation passes through the center of dilation.
You could also graphically illustrate this on a coordinate plane, using a ruler to draw the points and see if they align through a common center. If they do, it serves as visual confirmation.
Ultimately, if the computed coordinates yield the same center based on the transformations, then the answer is yes, confirming that the dilation passes through the center of dilation.
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