Let x be the length of the longer leg of the right triangle. Then the shorter leg is x - 14.
According to the Pythagorean Theorem, the length of the hypotenuse c is given by:
c^2 = a^2 + b^2
where a and b are the lengths of the legs of the right triangle.
Substituting the given values, we get:
c^2 = x^2 + (x-14)^2
Expanding and simplifying, we get:
c^2 = 2x^2 - 28x + 196
Since we know that the hypotenuse must be at least 26 cm, we can write:
c^2 >= 26^2
c^2 >= 676
Substituting the expression for c^2 from above, we get:
2x^2 - 28x + 196 >= 676
Simplifying and solving for x, we get:
2x^2 - 28x - 480 >= 0
x^2 - 14x - 240 >= 0
(x - 24)(x + 10) >= 0
The solutions to this inequality are x >= 24 or x <= -10. Since x represents a length, we can ignore the negative solution. Therefore, the smallest length of the longer leg is 24 cm.