To solve the problem:

Step 1: Understand the problem

• The figure shows two overlapping squares.
• The shaded region (area of overlap) is $51 \, \text{cm}^2$.
• The side length of the smaller square is $3 \, \text{cm}$.
• Let $x \, \text{cm}$ be the side length of the larger square.
• We need to find the areas of both squares.

Step 2: Set up the equation for the shaded region

The area of the shaded region is the difference between the area of the larger square and the area of the smaller square:

$\text{Area of shaded region} = \text{Area of larger square} - \text{Area of smaller square}$

Given: $51 = x^2 - 3^2$

Step 3: Solve for $x^2$

Substitute $3^2 = 9$:

$51 = x^2 - 9$

Add $9$ to both sides:

$x^2 = 51 + 9$

$x^2 = 60$

Step 4: Calculate the areas

1. The area of the larger square: $\text{Area of larger square} = x^2 = 60 \, \text{cm}^2$

2. The area of the smaller square: $\text{Area of smaller square} = 3^2 = 9 \, \text{cm}^2$

Final Answer:

• Area of the larger square: $60 \, \text{cm}^2$
• Area of the smaller square: $9 \, \text{cm}^2$

Would you like a detailed explanation of the solution steps, or help with further questions?

Here are 5 related questions to expand your understanding:

1. How would the solution change if the smaller square's side length were different?
2. Can the shaded area ever exceed the area of the smaller square?
3. What happens if both squares are centered?
4. How can we find the side length of the larger square from this solution?
5. Can this concept be applied to overlapping circles?

Tip: Always double-check your calculations when working with areas and geometric shapes!