Question

The Bermuda Triangle is roughly defined by Hamilton, Bermuda; San Juan, Puerto Rico; and Fort Lauderdale, Florida. The distances from Hamilton to Fort Lauderdale, Fort Lauderdale to San Juan, and San Juan to Hamilton are approximately \(1028,1046,\) and 965 miles, respectively. Ignoring the curvature of Earth, approximate the area of the Bermuda Triangle.

Short answer

The area of the Bermuda Triangle is approximately 446,700 square miles.

Step-by-step solution

- Identify the vertices and distances

The vertices of the Bermuda Triangle are Hamilton (H), San Juan (S), and Fort Lauderdale (F). The sides of the triangle are as follows: Hamilton to Fort Lauderdale (HF) is 1028 miles, Fort Lauderdale to San Juan (FS) is 1046 miles, and San Juan to Hamilton (SH) is 965 miles.

- Apply Heron's Formula

Use Heron's formula to calculate the area of the triangle. Heron's formula is \[A = \sqrt{s(s-a)(s-b)(s-c)}\], where \(a = 1028\), \(b = 1046\), \(c = 965\) miles are the sides of the triangle, and \(s\) is the semi-perimeter of the triangle.

- Calculate the semi-perimeter

First, find the semi-perimeter \(s\) using the formula \[s = \frac{a+b+c}{2}\]. Substitute the given values: \[s = \frac{1028 + 1046 + 965}{2} = 2019.5\] miles.

- Substitute into Heron's Formula

Substitute \(s\), \(a\), \(b\), and \(c\) into Heron's formula: \[A = \sqrt{2019.5 \times (2019.5 - 1028) \times (2019.5 - 1046) \times (2019.5 - 965)}\].

- Simplify the Expression

Calculate inside the square root: \[A = \sqrt{2019.5 \times 991.5 \times 973.5 \times 1054.5}\].

- Final Calculation

Use a calculator to find the final value: \[A \approx 446,700\] square miles.

Key concepts

semi-perimeter

The semi-perimeter of a triangle is a key concept in geometry, especially when using Heron's formula to compute the area. The semi-perimeter, denoted as \(s\), is calculated as half the sum of the lengths of the triangle's sides. This is a preliminary step in applying Heron's formula.

For a triangle with sides of lengths \(a\), \(b\), and \(c\), the semi-perimeter is given by:
\[ s = \frac{a + b + c}{2} \]

In the context of the Bermuda Triangle, where the sides are \(1028\) miles, \(1046\) miles, and \(965\) miles, we find the semi-perimeter as follows:
\[s = \frac{1028 + 1046 + 965}{2} = 2019.5\] miles.

Understanding the semi-perimeter is essential because it acts as a stepping stone to further calculations in finding the area using Heron's formula.

area of a triangle

Finding the area of a triangle can be done using various methods, but Heron's formula is particularly useful when we know the lengths of all three sides. This formula avoids the need for height measurements or angle calculations.

Heron's formula states:
\[A = \sqrt{s(s-a)(s-b)(s-c)}\]
where \(s\) is the semi-perimeter and \(a\), \(b\), and \(c\) are the sides of the triangle.

To apply this to the Bermuda Triangle:
\[s = 2019.5\] miles,
\[a = 1028\] miles,
\[b = 1046\] miles,
\[c = 965\] miles.

Substituting these values into the formula:
\[A = \sqrt{2019.5(2019.5 - 1028)(2019.5 - 1046)(2019.5 - 965)}\]

Simplifying inside the square root gives us:
\[A = \sqrt{2019.5 \times 991.5 \times 973.5 \times 1054.5}\]

Finally, using a calculator, we find:
\[A \approx 446,700\] square miles.

This detailed computation shows how the area of the triangle can be derived directly from the side lengths, thanks to Heron's formula.

distance calculation

Accurate distance calculation is crucial for solving geometric problems involving real-world data, like the Bermuda Triangle. In our example, the distances between the points Hamilton, San Juan, and Fort Lauderdale are approximately:

• Hamilton to Fort Lauderdale: \(1028\) miles
• Fort Lauderdale to San Juan: \(1046\) miles
• San Juan to Hamilton: \(965\) miles

These distances form the sides of our triangle. When calculating such measurements, it's essential to ensure precision as they influence the accuracy of subsequent calculations, like finding the semi-perimeter and area.

For students working with geographic data, ignoring the curvature of the Earth might simplify calculations but also introduces a bit of abstraction. However, this assumption allows us to treat the Earth's surface as a flat plane, making our distance calculations easier to manage.

Proper understanding and use of distance data are foundational steps in solving more complex problems in geometry and trigonometry.