Question

A right triangle has sides whose lengths are consecutive integers. Find the lengths of the sides.

Answer 1

3 cm, 4 cm, and 5 cm. How? The sum of 3x3 and 4x4 equals 5x5.

Answer 2

The 3,4,5 right triangle is so well-known you should immediately know the answer.
(Also, useful is the 5,12,13 right triangle)

But if you did not know it, you could use algebra to find the numbers:
call the smallest length n. Because the legs are consecutive, the other two are n+1, and n+2

pythagoras says
n^2 + (n+1)^2 = (n+2)^2 (the longest side is the hypotenuse)
expand out to get
n^2 + n^2 +2n + 1 = n^2 +4n + 4
simplify to get
n^2 -2n - 3 =0
factor to get
(n-3)(n+1) = 0
or
n=3, n= -1
we do not have negative lengths. so our smallest leg is 3, and the other two are 3+1,and 3+2
or, 3,4,5