One leg of a right triangle has a length of 77 m. the other sides have lengths that are consecutive integers. find these lengths.

Let x and x+1 represent the consecutive integers, then x is the other leg and x+1 is the hypotenuse (because the hypotenuse must be greater than its sides).

Use Pythagorean Theorem:
  a²   + b² = c²
(77)² + x² = (x+1)²
5929 + x² = x² + 2x + 1
5928        =        2x
2964        =          x

x+1 = 2965

The other leg is 2964 m and the hypotenuse is 2965 m