If a path or route is triangular in nature, then you can apply your understanding of angle and side relationships to find the part of the path that is missing.  Consider the Bermuda Triangle, the area in the Atlantic Ocean that has vertices ranging from the tip of Miami, Florida, to Bermuda to San Juan, Puerto Rico. It is in this region that many violent storms, shipwrecks or missing sea vessels, and other mysterious mishaps have occurred, which makes any route through the triangle a bit scary for some people. Still, the Bermuda Triangle remains one of the most heavily traveled routes, used by commercial vessels such as cruise ships and aircraft in addition to Naval and Coast Guard vessels. 

Other controversies surround the Bermuda Triangle, besides the arguments over the relative safety of routes through the area. One of those controversies involves size. Various researchers have determined that the actual Bermuda Triangle covers anywhere from 500,000 square miles to 1.5 million square miles. 

Suppose you have a chance to take a trip with to Puerto Rico with a friend. The problem is: Your friend is wary of traveling through or over the Bermuda Triangle. You know the location and distance for two sides of the Bermuda Triangle, but need to find the other side--so that you can reassure your friend that your trip will not cross the Bermuda triangle but will go around it. How can the Triangle Inequality Theorem help you? Click each step below to find out.

If you know the distance between Florida and Bermuda is 1,051 miles, and the distance from Puerto Rico to Bermuda is 979 miles, what should you do first to figure out the other side of the triangle--the distance between Florida and Puerta Rico?	Start by drawing the triangle FBP (F for Florida, B for Bermuda, and P for Puerto Rico). Then, label the sides of the triangle with the information you already have. Using the information you have, you can say that \(\small\mathsf{FB}\) = 1051 and \(\small\mathsf{PB}\) = 979.
What is your next step, using the Triangle Inequality Theorem?	Next you'll need to find the sum of the two sides you do know. \(\small\mathsf{FB}\) + \(\small\mathsf{PB}\) = 1051 + 979 2030 miles
What can you say about \(\small\mathsf{FP}\) (the distance between Florida and Puerto Rico) in relation to the other sides?	According to the Triangle Inequality Theorem, we can say that \(\small\mathsf{FP}\) < \(\small\mathsf{FB}\) + \(\small\mathsf{PB}\), or \(\small\mathsf{FP}\) < 2030. In other words, the distance between Florida and Puerto Rico must be less than 2,030 miles.
How can you get a little more precise about the length of the third side of the Bermuda Triangle? Try writing another inequality--this time involving the missing path, x, and \(\small\mathsf{PB}\).	According to the Triangle Inequality Theorem, we can also say that 1051 < x + \(\small\mathsf{PB}\). Since, \(\small\mathsf{PB}\) = 979, we have that 1051 < x + 979 and 72 < x or that x > 72.
What's the verdict? What can you tell your friend about the route to Puerto Rico?	The distance between Florida and Puerto Rico must be greater than 72 miles but less than 2030 miles. If your flight information indicates that your mileage is at least slightly more than 2030, you may be able to reassure your friend that the airline plans to use a curved route that does not cross the Bermuda Triangle. Notice that we didn't use the inequality involving the missing path, x, with the longest path 1051. This inequality 979 < 1051 + x, would have yielded -72 < x or x > -72. These numbers would not form a triangle according to this theorem and also, a triangle side cannot be negative.