Answer:
Step-by-step explanation:
You want the lengths of the sides of a right triangle such that differences between them are 3 cm.
The problem statement describes a right triangle whose side lengths are an arithmetic progression with a common difference of 3 cm.
The only right triangle whose legs form an arithmetic progression is the 3-4-5 right triangle, and multiples of those lengths. The differences in the 3-4-5 triangle are 1, so we want a triangle with sides 3 times as long:
short leg: 9 cm
long leg: 12 cm
hypotenuse: 15 cm
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Additional comment
If you're not familiar with the 3-4-5 right triangle and its arithmetic progression property, you could write an equation for the short side (x) using the Pythagorean relation:
x² +(x+3)² = (x+6)²
2x² +6x +9 = x² +12x +36
(x -3)² = 36 . . . . . . . . . subtract (x²+12x) and factor
x = 3 ±6 . . . . . take the square root
x = 9 . . . . . . . -3 is an extraneous solution
x +3 = 12
x +6 = 15
