Final answer:
The area of a triangle given its vertices is found using the formula involving the determinant of the matrix with those coordinates. The formula is A = ½|x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|.
Step-by-step explanation:
To determine the area of a triangle with vertices at A(x1, y1), B(x2, y2), and C(x3, y3), we can use the formula derived from the determinant of a matrix. The area (A) of the triangle is given by:
A = ½ |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
This formula is essentially the absolute value of half the determinant of the matrix that is formed by the coordinates of the vertices of the triangle.
Example Calculation
If we want to find the area of a triangle with vertices A(1,2), B(3,4), and C(5,6), we would substitute the values into the formula as follows:
A = ½ |1(4-6) + 3(6-2) + 5(2-4)|
= ½ |(-2) + (12) - (10)|
= ½ |0|
= 0
In this example, the area of the triangle is 0, which indicates that the points are collinear (all on the same line), and thus, do not form a triangle with a measurable area.