To show that the shaded figure (pre-image) is similar to the unshaded figure, we need to describe a sequence of transformations that maps one figure onto the other. Let's break it down based on the grid and the position of the points:

Step 1: Translation

• First, we can translate the shaded rectangle downwards to align it with the position of the unshaded rectangle. Specifically, we can move points $A'$, $B$, $C$, and $D$ to align with points $A$, $B$, $C'$, and $D'$.
• This translation is 4 units down vertically.

Step 2: Dilation (Scaling)

• After translating, the next transformation is dilation. We observe that the pre-image is half the width of the unshaded figure. To make the figures the same size, we can apply a dilation with a scale factor of 2.
• The center of dilation can be point $C$, as it remains fixed during the transformation.

By applying these two transformations—translation followed by dilation—the shaded pre-image will be mapped onto the unshaded figure, showing that they are similar. The figures have the same shape but different sizes, satisfying the condition of similarity.

Would you like further details or have any questions?

Here are 5 relative questions to expand on this:

1. What are the properties of similar figures in geometry?
2. How do we find the scale factor in a dilation transformation?
3. Can translation alone prove similarity between figures?
4. What other transformations could be used to prove similarity in certain cases?
5. How do you choose the center of dilation in similarity transformations?

Tip: When proving similarity, ensure that you always check if the sides are proportional and the angles are preserved.