The perimeter of a rectangle measures $100 \mathrm{~cm}$ and the diagonal measures $x \mathrm{~cm}$. What is the area of this rectangle as a function of $x$?
(a) $625-x^{2}$
(b) $625-\frac{x^{2}}{2}$
(c) $1250-\frac{x^{2}}{2}$
(d) $225-\frac{x^{2}}{2}$
(e) $2500-\frac{x^{2}}{2}$
The correct option is (c).

The area $A$ of a rectangle is the product of its length and width. Let $a$ and $b$ be the length and width of the rectangle. Thus, $A = ab$. The perimeter of this rectangle is given by $2a + 2b$. Since the perimeter is 100, we have $2a + 2b = 100$, therefore, $a + b = 50$. Squaring both sides of this last equality, we get

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-045.jpg?height=297&width=494&top_left_y=954&top_left_x=1341)
$a^2 + b^2 + 2ab = (a + b)^2 = 50^2 = 2500$. If $x$ denotes the length of the diagonal, the Pythagorean Theorem states that $x^2 = a^2 + b^2$, therefore, $x^2 + 2A = x^2 + 2ab = 2500$. We conclude that $2A = 2500 - x^2$, or $A = 1250 - \frac{1}{2} x^2$ is the expression for the area of the rectangle in terms of the diagonal $x$.