If a perimeter of a rectangle is 82 cm and its diagonal is 29 cm long, what is the area of the rectangle?

To solve this problem, we need to use the properties of a rectangle and the Pythagorean theorem. Here are the steps:

1. We know that the perimeter of a rectangle is 2*(length + width). So, 82 = 2*(length + width). This simplifies to length + width = 41.

2. We also know that the diagonal of a rectangle forms a right triangle with the length and width. So, we can use the Pythagorean theorem, which states that the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (length and width). So, 29^2 = length^2 + width^2.

3. Now we have a system of two equations: length + width = 41 and length^2 + width^2 = 29^2. We can solve this system to find the values of length and width.

4. To solve this system, we can square the first equation to get (length + width)^2 = 41^2. Expanding this gives length^2 + 2lengthwidth + width^2 = 41^2.

5. Subtract the second equation from this to get 2lengthwidth = 41^2 - 29^2. Solving for lengthwidth gives lengthwidth = (41^2 - 29^2) / 2.

6. Finally, the area of a rectangle is length * width, so the area is (41^2 - 29^2) / 2 = 420 cm^2.