How do I find a random point on the hypotenuse?

I'm trudging through 5-6 years worth of forgotten math lessons and can't remember how to do this. I have a right triangle, the two sides both are equal to 30 so the angles are a clean 45 degrees a piece. I need to know how to determine the x,y coords of a random point on the hypotenuse. I have a diagram but forgive it's crudeness, it's just an example:

http://www.innerfilth.com/%7Emike/distance.gif

Those red dots are what I need to figure out. Ignore their position, since it's just an example. If I pointed to a random spot on that line, what math do I need to figure out the coords?