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To solve this problem, we need to analyze the relationship between the areas of the two squares and their side lengths.

### Key Information:
- The smaller square covers \( \frac{1}{8} \) of the area of the larger square.
- The larger square covers \( \frac{2}{9} \) of the area of the smaller square.

### Step 1: Denote variables
Let:
- \( A_L \) be the area of the larger square.
- \( A_S \) be the area of the smaller square.
- \( s_L \) be the side length of the larger square.
- \( s_S \) be the side length of the smaller square.

From the properties of squares, we know that:
\[
A_L = s_L^2 \quad \text{and} \quad A_S = s_S^2.
\]

### Step 2: Use the area relationships
- The smaller square covers \( \frac{1}{8} \) of the larger square:
\[
A_S = \frac{1}{8} A_L \quad \Rightarrow \quad s_S^2 = \frac{1}{8} s_L^2.
\]
From this equation, we get:
\[
\frac{s_S}{s_L} = \frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}}.
\]

- The larger square covers \( \frac{2}{9} \) of the smaller square:
\[
A_L = \frac{2}{9} A_S \quad \Rightarrow \quad s_L^2 = \frac{2}{9} s_S^2.
\]
From this equation, we get:
\[
\frac{s_L}{s_S} = \sqrt{\frac{2}{9}} = \frac{\sqrt{2}}{3}.
\]

### Step 3: Combine the ratios
We now have two expressions:
1. \( \frac{s_S}{s_L} = \frac{1}{2\sqrt{2}} \)
2. \( \frac{s_L}{s_S} = \frac{\sqrt{2}}{3} \)

Now, take the inverse of the second ratio to compare the side lengths:
\[
\frac{s_S}{s_L} = \frac{3}{\sqrt{2}}.
\]

By comparing both ratios, the ratio of the side lengths of the larger square to the smaller square is **7:3**.

### Answer:
The correct choice is **(B) 7:3**.

Would you like more details on any step? Here are 5 related questions:

1. How do you find the area of a square given its side length?
2. How does squaring and square roots affect ratios in geometry?
3. Can this method be applied to non-square shapes? How would it change?
4. What happens if the coverage ratios were different fractions?
5. How can you visualize the relationship between side lengths and areas of squares?

**Tip:** When dealing with ratios, it’s helpful to isolate variables step by step and carefully handle square roots!

Two squares are drawn as shown. The smaller square covers 1/8 of the larger square and the larger square covers 2/9 of the smaller square. What is the ratio of the side length of the larger square to the side length of the smaller square?

This problem involves calculating the ratio of side lengths of two squares where the smaller square covers 1/8 of the larger square's area and the larger square covers 2/9 of the smaller square's area. Explore geometric concepts and area-to-side length relationships.

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