I know that the area of a triangle is half the product of its base and height, and that the base and height are perpendicular to each other. Since the triangle is inscribed in a circle, the hypotenuse must be the diameter of the circle, which is $100$ units long. To maximize the area of the triangle, I want to maximize the product of its base and height. I recall that for a fixed hypotenuse, the base and height are maximized when the triangle is isosceles, meaning that the base and height are equal. Therefore, I can let $x$ be the base and height of the triangle, and use the Pythagorean theorem to find $x$. I have $x^2 + x^2 = 100^2$, which simplifies to $2x^2 = 10000$. Dividing both sides by $2$, I get $x^2 = 5000$. Taking the positive square root of both sides, I get $x = \sqrt{5000}$. The area of the triangle is then half the product of $x$ and $x$, which is $\frac{1}{2}x^2$. Substituting $x = \sqrt{5000}$, I get $\frac{1}{2}(\sqrt{5000})^2$, which is $\frac{1}{2}(5000)$. Simplifying, I get $2500$. 
A right triangle is inscribed in a circle with a diameter $100$ units long. What is the maximum area of the triangle, in square units?