Calculate Triangle Area Using Coordinates: A Step-by-Step Guide

To find the area of a triangle using coordinates, follow these steps:

1. Identify the three vertices of the triangle and write down their coordinates. Let's call them A(x1,y1), B(x2,y2), and C(x3,y3).

2. Use the distance formula to find the length of each side of the triangle. The distance formula is:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

So, the length of AB is dAB = sqrt((x2 - x1)^2 + (y2 - y1)^2), the length of BC is dBC = sqrt((x3 - x2)^2 + (y3 - y2)^2), and the length of AC is dAC = sqrt((x3 - x1)^2 + (y3 - y1)^2).

3. Use Heron's formula to calculate the area of the triangle. Heron's formula is:

Area = sqrt(s(s-dAB)(s-dBC)(s-dAC))

where s is the semiperimeter of the triangle, which is half the perimeter:

s = (dAB + dBC + dAC) / 2

4. Plug in the values for dAB, dBC, and dAC, and calculate the area.

For example, let's say the coordinates of the vertices of a triangle are A(1,2), B(3,4), and C(5,1).

dAB = sqrt((3 - 1)^2 + (4 - 2)^2) = sqrt(8)

dBC = sqrt((5 - 3)^2 + (1 - 4)^2) = sqrt(13)

dAC = sqrt((5 - 1)^2 + (1 - 2)^2) = sqrt(17)

s = (sqrt(8) + sqrt(13) + sqrt(17)) / 2

Area = sqrt(s(s-sqrt(8))(s-sqrt(13))(s-sqrt(17))) = 5.5

Therefore, the area of the triangle with vertices A(1,2), B(3,4), and C(5,1) is 5.5 square units.