Calculating Rectangle Area from Square Perimeter Information

This problem involves finding the area of a rectangle given information about its relationship with a square's perimeter and its own dimensions. We need to calculate the side of the square first, then its perimeter. This perimeter will be equal to the rectangle's perimeter. Using the rectangle's perimeter and the relationship between its length and breadth, we can find its dimensions and then calculate its area.

Step 1: Find the Side of the Square

We are given the diagonal of the square, $d = 29\sqrt{2}$ cm. The relationship between the diagonal ($d$) and the side ($s$) of a square is given by the formula:

$$ d = s\sqrt{2} $$

Substituting the given diagonal:

$$ 29\sqrt{2} \text{ cm} = s\sqrt{2} $$

Dividing both sides by $\sqrt{2}$, we find the side of the square:

$$ s = 29 \text{ cm} $$

Step 2: Calculate the Perimeter of the Square

The perimeter of a square is calculated using the formula:

$$ \text{Perimeter}_{\text{square}} = 4s $$

Using the side length we found:

$$ \text{Perimeter}_{\text{square}} = 4 \times 29 \text{ cm} $$

$$ \text{Perimeter}_{\text{square}} = 116 \text{ cm} $$

Step 3: Relate Square Perimeter to Rectangle Perimeter

The problem states that the perimeter of the square is equal to the perimeter of the rectangle.

$$ \text{Perimeter}_{\text{rectangle}} = \text{Perimeter}_{\text{square}} = 116 \text{ cm} $$

Step 4: Find the Dimensions of the Rectangle

Let the breadth of the rectangle be $b$ cm.

The length ($l$) of the rectangle is 8 cm more than the breadth:

$$ l = b + 8 \text{ cm} $$

The formula for the perimeter of a rectangle is:

$$ \text{Perimeter}_{\text{rectangle}} = 2(l + b) $$

Substitute the known perimeter and the expression for length:

$$ 116 \text{ cm} = 2((b + 8) + b) $$

$$ 116 = 2(2b + 8) $$

$$ 116 = 4b + 16 $$

Subtract 16 from both sides:

$$ 116 - 16 = 4b $$

$$ 100 = 4b $$

Divide by 4 to find the breadth:

$$ b = \frac{100}{4} \text{ cm} $$

$$ b = 25 \text{ cm} $$

Now, find the length using $l = b + 8$:

$$ l = 25 + 8 \text{ cm} $$

$$ l = 33 \text{ cm} $$

Step 5: Calculate the Area of the Rectangle

The area of a rectangle is calculated using the formula:

$$ \text{Area}_{\text{rectangle}} = l \times b $$

Substitute the calculated length and breadth:

$$ \text{Area}_{\text{rectangle}} = 33 \text{ cm} \times 25 \text{ cm} $$

$$ \text{Area}_{\text{rectangle}} = 825 \text{ cm}^2 $$

The area of the rectangle is 825 square centimeters.