To find the area of quadrilateral (ABCD) with coordinates (A(1,2)), (B(3,5)), (C(7,6)), (D(5,3)), we use the shoelace formula for polygons.

Step 1: Recall the Shoelace Formula

For a polygon with vertices ((x_1,y_1), (x_2,y_2), ..., (x_n,yn)) (ordered clockwise/counter-clockwise), the area is:
[ \text{Area} = \frac{1}{2} \left| \sum{i=1}^n (xi y{i+1} - x_{i+1} yi) \right| ]
where (x{n+1}=x1) and (y{n+1}=y_1).

Step 2: Apply the Formula

List the coordinates in order, repeating the first at the end:
(A(1,2), B(3,5), C(7,6), D(5,3), A(1,2))

Calculate the sum of (xi y{i+1}):
[ (1×5) + (3×6) + (7×3) + (5×2) = 5 + 18 + 21 + 10 = 54 ]

Calculate the sum of (yi x{i+1}):
[ (2×3) + (5×7) + (6×5) + (3×1) = 6 + 35 + 30 + 3 = 74 ]

Compute the absolute difference and take half:
[ \text{Area} = \frac{1}{2} |54 - 74| = \frac{1}{2}×20 = 10 ]

Answer: (\boxed{10})

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