To solve the problem:

Step 1: Define Variables

Let:

• $x$ = side length of the larger square (in cm),
• $3 \, \text{cm}$ = side length of the smaller square.

The shaded region represents the area of the larger square minus the area of the smaller square.

Step 2: Write the Area Equation

The area of the larger square is: $x^2 \, \text{cm}^2.$ The area of the smaller square is: $3^2 = 9 \, \text{cm}^2.$ The shaded area is given as 51 cm²: $x^2 - 9 = 51.$

Step 3: Solve for $x^2$

$x^2 = 51 + 9 = 60.$

Step 4: Find the Side Length of the Larger Square

$x = \sqrt{60} \approx 7.75 \, \text{cm}.$

Step 5: Calculate the Areas

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

Final Answer:

• Area of the larger square: 60 cm²
• Area of the smaller square: 9 cm²

Would you like further clarification or a visual diagram?

Here are five related questions for deeper exploration:

1. How is the area formula for squares derived geometrically?
2. Can you generalize this problem for any square overlap configuration?
3. What are the implications if the smaller square is rotated or displaced?
4. How would the solution change if the shaded area was a fraction of the larger square’s area?
5. What is the significance of using perfect square roots in geometric calculations?

Tip: Always ensure dimensions are clearly labeled when solving geometry problems to avoid confusion!