Do two right triangles with the same length hypotenuse have the same area?

No. Consider the following picture:

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All the triangles have the same hypothenuse, but one of them has area of $r^2$ while others might have it arbitrarily small.

I hope this helps $\ddot\smile$

The answer is already given but I couldn't resist this explanation.

Imagine a ladder that is leaned against the wall and slowly sliding down the wall. At every time you will get a right triangle with the same hypotenuse. If we let $f(t)$ be the area at time $t$ then $f(t)$ will be continuous and at some time (when the ladder hits the floor) it will be zero. At previous times it is positive so by the intermediate value theorem it will achieve every value in some interval $[0, x]$ for some positive $x$.

No. With the hypothenuse as base, the area is the same if and only if the height is the same. Using Thales, you can see that there are points on the semicircle that have different distances from the hypothenuse/diameter.