To solve this problem, we need to approach it step by step:

1. Finding the side length of the smaller square:

We are given that the area of the smaller square formed by joining the midpoints of the larger square is 16 square meters. Let the side length of this smaller square be denoted as $y$.

The formula for the area of a square is: $\text{Area} = y^2$ Given that the area of the smaller square is 16 square meters: $y^2 = 16$ Taking the square root of both sides: $y = 4 \, \text{meters}$

2. Relating the side length of the larger square (x) to the smaller square:

The smaller square is formed by joining the midpoints of the sides of the larger square. This smaller square is rotated 45 degrees with respect to the larger square. The diagonal of the smaller square corresponds to the side length of the larger square.

The relationship between the side length $y$ of a square and its diagonal $d$ is given by the formula: $d = y \sqrt{2}$ In our case, the diagonal $d$ of the smaller square is equal to the side length $x$ of the larger square: $x = 4\sqrt{2}$ Thus, the side length of the larger square is: $x = 4\sqrt{2} \approx 5.656 \, \text{meters}$

3. Finding the area of the larger square:

The area $A$ of the larger square is given by: $A = x^2 = (4\sqrt{2})^2 = 16 \times 2 = 32 \, \text{square meters}$

4. Finding the area outside the smaller square:

The area of the portion of the larger square that is outside the smaller square is simply the difference between the area of the larger square and the area of the smaller square: $\text{Area outside the smaller square} = 32 - 16 = 16 \, \text{square meters}$

Final Answer:

• The side length of the larger square $x$ is $4\sqrt{2}$ meters (approximately 5.656 meters).
• The area of the portion of the larger square that is outside the smaller square is 16 square meters.

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Would you like any more details or clarifications?

Here are 5 related questions to explore further:

1. What is the diagonal of the larger square?
2. How would this problem change if the smaller square’s area were given as 25 square meters?
3. Can we generalize this problem for any given smaller square area?
4. How does the relationship between a square's diagonal and side length help solve this type of problem?
5. How would the result differ if the smaller shape formed was a rectangle instead of a square?

Tip: When solving geometry problems involving squares, pay attention to the relationship between the diagonal and side length. This relationship often simplifies complex calculations.