Question 973973
The hypotenuse, c,  is sqrt(8100)=90

You draw the altitude, and that makes two triangles, each of which is similar to the original.  I would draw all three triangles in the same way to show similarity and label them accordingly.  

The whole hypotenuse is to the shorter leg, 54, as the shorter leg, 54, is to the shorter leg of the bisected hypotenuse.

(90/54)=(54/x)
2916=90x; x=32.4

(a/54)=(54/90) also works.
You don't have to solve for b, although it is easy enough.

a^2 +h^2=54^2
(32.4^2)+ h^2 = 2916
1049.76 +h^2 =2916
h^2=1866.24
h=43.2 units ANSWER

check with b
(43.2)^2 + 57.6^2=72^2
1866.24 + 3317.76= 5084