Question

A tree is supported by a wire anchored in the ground 5 feet from its base. The wire is 1 foot longer than the height that it reaches on the tree. Find the length of the wire

Answer 1

A Pythagorean triple with a short side of 5 and other lengths differing by 1 is the triple (5, 12, 13).

The wire is 13 feet long.

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If you want to work it out algebraically, you can let x represent the length of the wire. Then the Pythagorean theorem tells you
  x^2 = (x -1)^2 +5^2
  x^2 = x^2 -2x +1 +25
  2x = 26
  x = 13
The length of the wire is 13 feet.