the altitude to the hypotenuse of a right triangle divides the hypotenuse into segments of 9cm and 20cm. What is the length, of he shorter l

We have that

h/sin(theta) = 20/ cos(theta) →  h = 20 tan(theta)

And

h/cos(theta) = 9/ sin(theta)   → h = 9cot(theta)

So

20tan(theta)  = 9cot (theta) →

tan^2(theta)  = 9/20

tan(theta) = (2/3)√5

Therefore

tan-1( (2/3)√5) = theta = about 56.15°

So the shorter side is given by

cos(56.15) = 9 / S      where S is the shorter side

S = 9/cos(56.15) = 16.155

And the longer side is given by

cos(90- 56.15) = 20/ L

L = 20/cos(33.84) =  24.08

Check

√[16.155^2 + 24.08^2] ≈ 29

Since the smaller interior triangle is similar to the full triangle, this proportion can be created (where x is the length of the shorter leg):

9/x  = x/29

Cross multiply:  x·x  =  9·29     --->   x²  =  261     --->   x = √261