Question

Which set of lengths could be the side lengths of a right triangle?

20 cm, 22 cm, 29 cm
10 ft, 12 ft, 15 ft
7 km, 24 km, 28 km
13 in., 84 in., 85 in.

Answer 1

5,12,13 is a right triangle. it's one of the pythagorean triples.
there are many more. pythagorean triples are a set of three numbers that are lengths of the sides of a right triangle. the set of numbers are ratios.
3,4,5
8,15,17
5,12,12
7,24,25
9,40,41
11, 60, 61
13, 84, 85
16, 63, 65
20, 21, 29
28, 45, 53
33, 56, 65
36, 77, 85
39, 80, 89
48, 55, 73
65, 72, 97
12, 35, 37

Answer 2

For a right triangle:
c^2 = a^2 + b^2
Therefore:
12^2 + 9^2 = 15^2
4^2 + 14^2 = 212 ≠ 12^2
5^2 + 12^2 = 169 = 13^2