Area of a Triangle math

Find the area of the triangle T in the plane with vertices A(2,4), B(3,7), and C(4,5).

The area of a triangle can be given in terms of its coordinates by the determinant formula:

| | x1 y1 1 |
| | x2 y2 1 | x 1/2 |
| | x3 y3 1 |

In other words, we take the three coordinate pairs, append 1 to each one and place them in the rows of a matrix, take the determinant, multiply by 1/2 and then take the absolute value of the result.

The area in this case comes out to

| | 2 4 1 |
| | 3 7 1 | x 1/2 |
| | 4 5 1 |

| (2 x 7 x 1 + 4 x 1 x 4 + 1 x 3 x 5 - 1 x 7 x 4 - 2 x 1 x 5 - 4 x 3 x 1 ) x 1/2 |
= | (14 + 16 + 15 - 28 - 10 - 12) x 1/2 |
= | (45 - 50) x 1/2 |
= | (-5) x 1/2 |
= | -5/2 |
= 5/2 (or 2 1/2 or 2.5)