Answer
The lengths are 14 cm and 15 cm.
Work Step by Step
We call one length $x$ and the other length $29 - x$. Thus, the Pythagorean theorem for the entire triangle becomes:
$(x+6)^2 + (35-x)^2 = 29^2 \\ 2x^2 -58x +420 = 0 \\ (2)(x-14)(x-15) = 0 \\ x = 14, 15$
