The length of the longer leg of a right triangle is 6cm more than twice the length of the shorter leg. the length of the hypotenuse is 9cm more than twice the length of the shorter leg. find the side lengths of the triangle.

Step by step:

short = s,

long = 6 + 2*s,

hypotenuse = 9 + 2*s

Pythagoras:

(9+2s) ^2 = s^2 + (6+2s) ^2

81 + 36*s + 4*s^2 = s^2 + 36 + 24*s + 4*s^2

0 = s^2 - 12*s-45

s = - 3 or s = 15

Negative s, no sense.

So s=15, l=36, h = 39

Indeed: 39^2=36^2+15^2

Nice problem!