let 'a' and 'b' are the legs and a>b, and 'c' the hypotenuse, and a,b,c >0

one leg of a triangle is 1 cm longer than the other leg

a = b + 1

the length of the hypotenuse is 9 cm longer than that of the shorter leg

c = 9 + b

using the Pythagorean Theorem we have

a^2 + b^2 = c^2

plug a = b + 1 and c = 9 + b into the last equations

(b + 1)^2 + b^2 = (9 + b)^2

by solving we find

b = 20 cm

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a = 20 + 1

a = 21 cm

c = 20 + 9

c = 29 cm

the length of the 3 legs are: 21 cm, 20 cm and 29 cm.