Area of Bermuda Triangle

Date: 06/10/2001 at 21:06:10
From: Bronwyn Smith
Subject: Area of Triangles

The Bermuda triangle is shown on a graph with the points A(1,2) B(4,8) 
and C(8,1). Determine the area of the triangle two different ways.

I determined the area of the triangle by first finding the lengths of 
all the sides, then determining the median of the shortest length, 
where I continued to find the length from the median to the opposite 
angle. I then used half of the base times the height to figure out the 
area of the triangle. However, what other ways can you figure out an 
area of a triangle?

Date: 06/11/2001 at 12:59:03
From: Doctor Peterson
Subject: Re: Area of Triangles

Hi, Bronwyn.

I can think of several ways you could do this.

1. Use the distance formula to find the lengths of the sides, and use 
Heron's formula for the area (which you can find in our archives). 
This might involve very hard arithmetic.

2. Use the formula given in our FAQ (under Formulas, then Analytic 
Geometry) for the area of a polygon given the coordinates of the 
vertices.

3. Use the method by which that formula can be derived: Find the area 
of the three trapezoids formed by one of the edges of the triangles, 
the x-axis, and the vertical lines through the two end points. Then 
combine those three areas appropriately to find the area of the 
triangle.

4. Enclose the triangle in a rectangle, and subtract from its area the 
areas of the three right triangles that are left when you remove the 
given triangle from it.

5. Split the triangle into several smaller triangles and add their 
areas.

6. Use Pick's Rule to find the area of the triangle by counting 
lattice points on and in the triangle. (You can find this, too, in our 
archives.)

7. Find the point where one side intersects the perpendicular from the 
opposite vertex, and use that to find the altitude. (I think this 
might be what you meant to do; your "median" won't work because the 
altitude must be perpendicular to the base.)

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/