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### Determinants and the Area of a Triangle

```
Date: 12/14/98 at 13:50:41
From: Frank Chiaravalli
Subject: Matrices and determinants

The area of a triangle having vertices (A,B), (C,D), and (E,F) is the
absolute value of the determinant of M, where:

| A B 1 |
M = 1/2 | C D 1 |
| E F 1 |

How did the textbook arrive at this formula?

Many thanks for any help you can give us. Quite a few students have
```

```
Date: 12/14/98 at 14:41:59
From: Doctor Anthony
Subject: Re: Matrices and determinants

Draw a figure with vertices (A,B), (C,D), (E,F) in the first quadrant.
For the sake of argument let (A,B) be nearest the y axis, (C,D)
farthest from the y axis and (E,F) between the other two vertices and
lower than either so that it is closest to the x axis.

Now draw verticals from the vertices to the x axis:

The area of the triangle is found by finding the area of the largest
trapezium (that with one boundary: the line joining (A,B) to (C,D)) and
then subtracting two smaller trapezia, those with other two sides of
the triangle as boundaries.

The large trapezium has area (1/2)(B+D)(C-A), and the smaller trapezia
on the area of trapezia, see:

So the area of the triangle is:

(1/2)[(B+D)(C-A) - (B+F)(E-A) - (F+D)(C-E)]

(1/2)[BC-BA+DC-DA - BE+BA-FE+FA - FC+FE-DC+DE]

(1/2)[BC - DA - BE + FA - FC + DE]

(1/2)[-AD - BE - CF + ED + FA + BC]

compared with:

|A  B  1|
(1/2)|C  D  1|  = (1/2)[AD + BE + CF - ED - FA - BC]
|E  F  1|

and apart from being opposite in sign the two expressions are the same.
So the determinant gives twice the area of the triangle.

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Coordinate Plane Geometry
High School Geometry
High School Linear Algebra
High School Triangles and Other Polygons

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