Is it possible for the lengths of a right triangle to form a geometric sequence? If yes, determine the lengths and the measures of the acute angles. if it's not possible, explain why not.

lengths of a right triangle? You mean the three sides?

Try it: lengths a, ar, ar^2

now see if Pythagoras postulate will work:

c^2=a^2 + b^2

Either a, or ar^2 is c. Try a first

a^2=(ar)^2+(ar^2)^2
dividing by a^2
1=r^2 + r^4
r^4+r^2-1=0

r^2 = (1+-sqrt(5))/2
r= 1.272
so sides are a; 1.272a; 1.618a

which means a was not the longest side.

check:
(1.618^2)=1^2+1.272^2

acute angles?
sinA= 1.272/1.618 Then B: sinB=1/1.618