To solve the problem, let's break it into steps:

Problem Information

1. Two squares overlap, and the shaded region is the difference in areas of the larger and smaller square, given as 51 cm².
2. The side length of the larger square is $x$ cm.
3. The smaller square has a side length of $x - 3$ cm.

We need to find:

• The area of the smaller square.
• The area of the larger square.

Step 1: Write the formula for the areas

1. Area of the larger square: $x^2$.
2. Area of the smaller square: $(x - 3)^2$.

Step 2: Use the given condition

The shaded region is the difference in the areas of the two squares: $x^2 - (x - 3)^2 = 51$

Step 3: Expand $(x - 3)^2$

$(x - 3)^2 = x^2 - 6x + 9$

Substitute into the equation: $x^2 - [x^2 - 6x + 9] = 51$

Simplify: $x^2 - x^2 + 6x - 9 = 51$

$6x - 9 = 51$

Step 4: Solve for $x$

$6x = 51 + 9$ $6x = 60$ $x = 10$

Step 5: Calculate the areas

1. Larger square's area: $x^2 = 10^2 = 100 \, \text{cm}^2$
2. Smaller square's area: $(x - 3)^2 = (10 - 3)^2 = 7^2 = 49 \, \text{cm}^2$

Final Answer:

• The area of the larger square is 100 cm².
• The area of the smaller square is 49 cm².

Would you like further details or clarifications?

5 Related Questions:

1. What would the areas be if the side length of the larger square was increased by 2 cm?
2. How is the formula for the area of a square derived?
3. If the shaded region was increased to 60 cm², how would the calculations change?
4. What happens to the shaded area if both squares' side lengths are doubled?
5. Can you calculate the side lengths if only the areas of the squares are given?

Tip:

Always simplify the equation step by step and double-check for algebraic errors when solving for unknowns in geometric problems!