Final answer:
The area of a larger square with twice the side length of a smaller square is four times greater. This is due to the squared relationship when scaling the dimensions of similar geometric figures such as squares.
Step-by-step explanation:
The student's question is about finding the area of a shaded square in a composite figure composed of congruent squares. From the question's details, we can investigate a similar principle by using Marta's example. Marta has a smaller square with a side length of 4 inches. If we have a larger square with side lengths that are twice that of the first square, the side length would be 8 inches (4 inches x 2).
The area of the larger square will compare to the smaller square such that it will be four times as large. This phenomenon occurs because the ratio of the areas of similar figures is the square of the scale factor.
When we scale a figure by a certain factor, the two-dimensional area scales by the square of that factor. For example, if the side length of a square is doubled, its area increases by a factor of four (2 squared). Thus, comparing Marta's smaller square with an area of 16 square inches (4 inches x 4 inches), the larger square's area is 64 square inches (8 inches x 8 inches), which confirms the squared relationship.