The Lengths Of The Sides Of A Triangle Are Consecutive Integers. Half Of The Perimeter Is 14 More Than The Length Of The Longest Side. Find The Perimeter?

Suppose s is the length of the middle-length side. The perimeter will be 3s, and the longest side will be (s+1). The problem statement says
  (3s)/2 = 14 + (s+1)
  (3/2)s = 15 + s    (collect terms on the right)
  (1/2)s = 15    (subtract s from both sides)
  s = 30    (multiply both sides by 2)
The perimeter is 3s = 90.