MHB The area of a triangle and determinants

Click For Summary
SUMMARY

The area of a triangle can be calculated using determinants formed by the triangle's vertices. Specifically, the determinant yields double the area when the triangle is oriented counter-clockwise. If the triangle is oriented clockwise, the result will be the negative of the actual area, explaining the discrepancy in signs noted in the discussion. This relationship is crucial for correctly applying determinants in geometric calculations.

PREREQUISITES
  • Understanding of determinants in linear algebra
  • Familiarity with triangle geometry and area calculation
  • Knowledge of coordinate systems and orientation
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of determinants in linear algebra
  • Learn how to calculate the area of a triangle using the determinant method
  • Explore the implications of triangle orientation on area calculations
  • Investigate applications of determinants in geometry and physics
USEFUL FOR

Mathematicians, geometry enthusiasts, students studying linear algebra, and anyone interested in the applications of determinants in geometric contexts.

Yankel
Messages
390
Reaction score
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: 125
  • 2.PNG
    2.PNG
    17.7 KB · Views: 133
  • 3.PNG
    3.PNG
    4.9 KB · Views: 122
  • 4.PNG
    4.PNG
    7.1 KB · Views: 130
Mathematics news on Phys.org
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.
 

Similar threads

MHB What is the area of triangle $STV$?
  • · Replies 1 ·
Replies
1
Views
1K
MHB Area of Triangle Shaded Region
  • · Replies 5 ·
Replies
5
Views
2K
MHB Prove Area of Equilateral Triangle
  • · Replies 9 ·
Replies
9
Views
2K
MHB Calculate Area of Triangle ABC
  • · Replies 2 ·
Replies
2
Views
5K
Comp Sci Computing the Length & Area of a 3D Triangle
  • · Replies 1 ·
Replies
1
Views
2K
MHB Equilateral triangle within a circumscribed circle
  • · Replies 5 ·
Replies
5
Views
3K
MHB Prove Equal Area Triangle & Parallelogram, $\angle ADC=90^{\circ}$
  • · Replies 9 ·
Replies
9
Views
4K
MHB How Can You Calculate the Area of a Polygon Using Coordinates?
  • · Replies 3 ·
Replies
3
Views
2K
Find the third coordinate of a vertex of an equilateral triangle
  • · Replies 11 ·
Replies
11
Views
10K
MHB Maximize Trapezoid Area with 3 Equal Sides | Leprofece Answer
  • · Replies 1 ·
Replies
1
Views
3K