Which of the following shows the length of the third side, in inches, of the triangle below? hypotenuse is 74 and one leg is 24

The Pythagorean theorem applies. If the sides are designated by "a" and "b", and the hypotenuse is designated by "c", then that theorem tells you
   c² = a² + b²

Filling in your numbers, we can solve for b.
   74² = 24² + b²
   74² - 24² = b² = 4900
   b = √4900 = 70

The length of the third side is 70.