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Have you ever wondered how transformations like dilation and rotation can bring out hidden relationships in geometric shapes? Today, we're diving into the fascinating world of quadrilaterals to uncover a mystery that lies within their sides.
Lucas, our inquisitive mathematician, has dilated quadrilateral ABCD to create a new quadrilateral, WXYZ. It appears he's rotated and zoomed in on the original shape, expanding it to create this new figure. But what does this tell us about their relationship?
We're presented with a challenge: determining which proportion must be true based on these transformations. Before we delve into the details, pause for a moment and see if you can figure it out on your own. What proportion comes to mind when you consider the similarity between these two quadrilaterals?
Now, let's explore together. By coloring the corresponding sides the same, we can easily identify the relationships between them. Side CD corresponds to side YZ, and the journey from angle C to angle Y mirrors the journey from angle A to angle W. These are the corresponding sides we'll focus on.
As we analyze the given ratios, we notice something peculiar. The ratio of CD to BC seems out of place. It should be the pink side to the blue side, not the other way around. This discrepancy leads us to eliminate one of the options.
Next, we encounter a ratio that doesn't even use corresponding sides. This can't be the correct proportion, so we rule it out as well.
Then, we come across the ratio of CD to YZ, which seems promising. However, we quickly realize that WZ does not correspond to BC, so this option is eliminated too.
Through a process of elimination and deductive reasoning, we arrive at the correct proportion. The ratio of CD to YZ is indeed the same as the ratio of BC to XY. This is the proportion that holds true for these similar quadrilaterals.
In conclusion, the journey through dilation and rotation has revealed a hidden truth about quadrilaterals. By understanding the relationships between corresponding sides, we can unlock the secrets of these shapes and their transformations.
So, the next time you encounter a geometric puzzle, remember to look for the corresponding sides and their ratios. You might just find the answer you're looking for.
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