The area of a triangle and determinants

In summary, the conversation discusses the attempt to prove that the area of a triangle is equal to the determinant of its three points. The person encountered an issue with the signs being incorrect in their proof. They share pictures and ask for clarification on how the two formulas can be identical when one comes from trapezoid subtraction and the other from the determinant. It is clarified that the determinant gives double the area of a counter-clockwise triangle, but in this case the triangle is oriented clockwise, resulting in the opposite area.
  • #1
Yankel
395
0
Dear all,

I was trying to prove that the area of a triangle is equal to the determinant consisting of the three points of the triangle. I got to the end, and something ain't working out. The signs are all wrong.

In the attached pictures I include my proof. Can you please tell me how can the two formulas be identical ? The first is the area coming from trapezoid subtraction , while the second is the determinant.

Thank you !

Clarification: when I say signs are opposite, I mean (y2-y3) vs. (y3-y2) , etc...

View attachment 7656

View attachment 7657

View attachment 7658

View attachment 7659
 

Attachments

  • 1.PNG
    1.PNG
    28.4 KB · Views: 78
  • 2.PNG
    2.PNG
    17.7 KB · Views: 82
  • 3.PNG
    3.PNG
    4.9 KB · Views: 75
  • 4.PNG
    4.PNG
    7.1 KB · Views: 82
Mathematics news on Phys.org
  • #2
Hi Yankel,

The determinant gives double the area of a triangle that is oriented counter-clockwise.
In your case the triangle is oriented clockwise, meaning that we'll find the opposite.
 

FAQ: The area of a triangle and determinants

What is the formula for finding the area of a triangle using determinants?

The formula for finding the area of a triangle using determinants is:
A = 1/2 * |x1 y1 1|
          |x2 y2 1|
          |x3 y3 1|
where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the triangle's vertices.

How does using determinants to find the area of a triangle differ from using the traditional formula (1/2 * base * height)?

Using determinants to find the area of a triangle is a more general method that can be applied to any type of triangle, while the traditional formula only works for right triangles. Additionally, the determinant formula does not require the use of a specific "base" and "height" like the traditional formula does.

What are the advantages of using determinants to find the area of a triangle?

One advantage of using determinants is that it can be applied to any type of triangle, including equilateral, scalene, and isosceles triangles. It also does not require the use of trigonometric functions, making it simpler and more efficient to use in certain situations.

Can determinants be used to find the area of a triangle in 3-dimensional space?

Yes, determinants can be used to find the area of a triangle in 3-dimensional space by adding a third coordinate (z) to each vertex and then using the formula:
A = 1/2 * |x1 y1 z1 1|
          |x2 y2 z2 1|
          |x3 y3 z3 1|
However, in 3-dimensional space, the result will be the area of the triangle's parallelogram, not the triangle itself.

How is the determinant of a triangle related to its area?

The absolute value of the determinant of a triangle is equal to twice the area of the triangle. This means that the area of the triangle can be calculated by taking half of the determinant's absolute value.

Similar threads

MHB Proving Triangle Area ≤ $\frac{1}{2}$ in a Square with $(n+1)^2$ Points
Replies
1
Views
972
MHB What is the area of triangle $STV$?
Replies
1
Views
1K
MHB Area of Triangle Shaded Region
Replies
5
Views
1K
MHB Prove Area of Equilateral Triangle
Replies
9
Views
2K
MHB Calculate Area of Triangle ABC
Replies
2
Views
5K
Comp Sci Computing the Length & Area of a 3D Triangle
Replies
1
Views
1K
MHB Equilateral triangle within a circumscribed circle
Replies
5
Views
3K
MHB Prove Equal Area Triangle & Parallelogram, $\angle ADC=90^{\circ}$
Replies
9
Views
3K
MHB How Can You Calculate the Area of a Polygon Using Coordinates?
Replies
4
Views
2K
Find the third coordinate of a vertex of an equilateral triangle
Replies
11
Views
9K

Hot Threads

Recent Insights

Back
Top