If I have a right angle triangle and I know the x and y dimentions, how do I calculate the height from the base to the slope along a certain point on the slope?

You can calculate the degree of the angle with sin/cos, and you can use this knowledge to calculate the height of the "new" triangle - assuming you know one of the other dimensions.

Only if you know both x and y of "new" triangle, but then why bother with the "old" triangle at all?

Yeah, misread that. If he has new x and y dimensions i don't think he can. He'd only have the angle and thats not enough.

Plot the hypotenuse as a function of the X coordinate. It will be of the form Y=mX, where m is the slope, which is height/base. Now your equation will give you the height (Y value) for any given distance (X value).

If I understand your question correctly, if you pick a point on the base and go straight up until you reach the hypotenuse you will create a new right triangle that has the same relationship as your original triangle. So if you know what the length of your new base is you can take it as a fraction of your original base and use that fraction to determine the height of your new triangle. For instance, say you have a 3-4-5 right triangle so that the lengths of the three sides are 3, 4, and 5. If I assume that the the side of length 3 is the base and I use a new base that is one half the length of the original I have a new base of 1.5. I can use this base and the original hypotenuse to create a new right triangle that has lengths of 1.5-2-2.5.