The Bermuda Triangle is a region of the Atlantic Ocean whose boundary is made up of straight lines connecting Bermuda, Florida, and Puerto Rico. Given that Florida is 1030 miles from Bermuda in the direction 24 degrees south of west and Puerto Rico is 980 miles from Bermuda in the direction four degrees west of south, find the area of the Bermuda Triangle.
There’s rather a lot of information in this question. So let’s begin by sketching out a diagram to help us understand what we’re going to need to do. We’re initially told that Florida is 1030 miles from Bermuda. And we’re told the direction. It’s 24 degrees south of west. We’ll first add the first vertex of this triangle. That’s Bermuda. And we’ll also add the four compass points, north, east, south, and west.
To find the direction that Florida lies, we start west. And we move south by 24 degrees as shown. And we know that Florida lies 1030 miles in this direction. Next, we’re told that Puerto Rico is 980 miles. And that’s in the direction four degrees west of south from Bermuda. This time, we start measuring from south and move four degrees in the direction of west. Then, we can see that Puerto Rico lies 980 miles in this direction.
And finally, we join the points representing Florida and Puerto Rico to complete our triangle. Then, to find the area of the triangle, we recall this formula. It’s a half 𝑎𝑏 sin 𝑐. We’re going to relabel our triangle so that it matches the letters in our formula. Let’s call the vertex representing Bermuda 𝐶, since we’re going to be able to work out the angle at this point. We can then label the vertices representing Florida and Puerto Rico, 𝐴 and 𝐵, in any order.
This means the side directly opposite the vertex 𝐶 is lowercase 𝑐. The side directly opposite vertex 𝐴 is lowercase 𝑎. And the side directly opposite vertex 𝐵 is lowercase 𝑏. So how do we work out the measure of the angle at 𝑐? Well, we can spot that the angle between the two compass points, south and west, must be 90 degrees. So we can say that the measure of the angle at 𝐶 plus this angle, 24 degrees, and this angle, four degrees, must add to make 90 degrees. In fact, 24 plus four is 28. So we can say that the measure of the angle at 𝐶 plus 28 degrees must be equal to 90 degrees.
We can solve this equation by subtracting 28 from both sides. And since 90 minus 28 is 62, we can see that the measure of the angle at 𝐶 is 62 degrees. And with that, we now have everything we need to be able to calculate the area of this triangle. 𝑎 is 980 and 𝑏 is 1030. So the area is a half multiplied by 980 multiplied by 1030 multiplied by sin of 62 degrees. And if we put that into our calculator, we get 445623.650 and so on. Correct to two decimal places, we can say that the area of the Bermuda Triangle is 445623.65 square miles. And an alternative way of representing square miles is as shown.