Find the area of the quadrilateral ABCD, if the coordinates of its vertices are known: A (1; 3), B (2; 6), C (4; 3), D (2; 1).

The area of an arbitrary quadrangle, the vertices of which are specified by the coordinates (x1; y1), (x2; y2), (x3; y3), (x4; y4), can be found by the formula:
S = (| (x1 – x2) (y1 + y2) + (x2 – x3) (y2 + y3) + (x3 – x4) (y3 + y4) + (x4 – x1) (y4 + y1) |) / 2.
Substitute the data on the condition of the coordinate (A (1; 3), B (2; 6), C (4; 3), D (2; 1)) into the formula and find the area of the quadrilateral ABCD:
S = (| (1 – 2) (3 + 6) + (2 – 4) (6 + 3) + (4 – 2) (3 + 1) + (2 – 1) (1 + 3) |) / 2 = (| (- 1) * 9 + (- 2) * 9 + 2 * 4 + 1 * 4 |) / 2 = (| – 9 – 18 + 8 + 4 |) / 2 = (| – 27 + 12 |) / 2 = | – 15 | / 2 = 15/2 = 7.5.
Answer: S = 7.5.