To determine which algebraic representation shows the dilation of trapezoid $ABCD$ to $A'B'C'D'$, we need to examine the scale factor of the transformation.

1. Identify the transformation type: Dilation involves scaling the coordinates of each point by a certain factor.

2. Measure or observe the scaling factor: From the diagram, it appears that trapezoid $A'B'C'D'$ is larger than trapezoid $ABCD$. We need to verify the scale factor by comparing the coordinates or observing proportional changes.

3. Analyze options based on scaling factors:

   • Option (x, y) → (2x, 2y) suggests a scale factor of 2 (each coordinate is doubled).
   • Option (x, y) → (3x, 2y) is an inconsistent scaling and does not represent a typical dilation.
   • Option (x, y) → (0.5x, 0.5y) suggests a reduction by half, or a scale factor of 0.5.
   • Option (2x, 2y) → (x, y) suggests a reduction by half in reverse.

Given the layout and alignment of the points in $A'B'C'D'$ relative to $ABCD$, the most likely correct answer is (x, y) → (2x, 2y), representing a dilation with a scale factor of 2.

Would you like further details on the solution, or do you have any questions?

Here are 5 related questions that might interest you:

1. How do you calculate the scale factor of a dilation?
2. What is the effect of different scale factors on geometric shapes?
3. How do dilations affect the area and perimeter of a shape?
4. Can dilations be represented with negative scale factors?
5. How does dilation differ from translation or rotation in transformations?

Tip: When identifying transformations, focus on whether coordinates are proportionally scaled, shifted, or rotated.