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The problem asks to find the area of square ACEG.

Looking at the diagram, square ACEG is inscribed inside another square, with points A, C, E, and G lying on the midpoints of the sides of the outer square (labeled B, D, F, and H). The outer square has its side lengths labeled as 8 units, so each side of the outer square is 8 units.

The vertices of square ACEG divide each side of the outer square into two equal parts, meaning each segment on the sides of the outer square is 4 units long. Therefore, square ACEG can be considered a smaller square rotated within the larger square.

To find the area of square ACEG:
1. First, calculate the diagonal of square ACEG. The diagonal is equal to the side of the outer square, which is 8 units.
2. For any square, the relation between the side \(s\) and the diagonal \(d\) is given by the formula:
   \[
   d = s \sqrt{2}
   \]
   Solving for \(s\):
   \[
   s = \frac{d}{\sqrt{2}} = \frac{8}{\sqrt{2}} = 8 \times \frac{\sqrt{2}}{2} = 4\sqrt{2} \text{ units}
   \]
3. Now, calculate the area \(A\) of square ACEG using the formula for the area of a square \(A = s^2\):
   \[
   A = (4\sqrt{2})^2 = 16 \times 2 = 32 \text{ square units}
   \]

Thus, the area of square ACEG is **32 square units**.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How do you find the diagonal of a square given its area?
2. What is the relationship between the diagonal and side length of a square?
3. Can you derive the area of square ACEG using a different method, such as coordinate geometry?
4. How would the area change if the outer square's side length was increased or decreased?
5. How does the orientation of the inner square affect the calculation of its area?

**Tip:** When solving problems involving inscribed shapes, look for symmetry or relationships between the shapes to simplify calculations.

Find the area of square ACEG. The square is inscribed inside a larger square with sides of length 8 units.

Find the area of square ACEG, which is inscribed in a larger square with side lengths of 8 units. This solution uses the formula for the diagonal of a square and basic geometry to calculate the area as 32 square units. Suitable for students in Grades 9-12.

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