Find the area of quadrilateral ABCD in which AB = 7 cm, BC = 6 cm, CD = 12 cm, DA = 15 cm, and AC = 9 cm.

The figure is given below:

As 12, 9, and 15 are Pythagorean triplets, Δ ACD is a right-angled Triangle.

(If a, b, and c are the sides of a triangle, Pythagorean triplet means, b² = a² + c²)

Area od Quadrilateral ABCD = Area of ΔACD + Area of ΔABC

Area of ΔACD = Area of the right-angled triangle

Area of ΔACD = 54 cm²

For finding Area of ΔABC we will use Heron’s Formula.

According to Heron’s Formula,

Where s = semiperimeter

a, b, and c are the sides of the triangle.

Perimeter of ΔABC = 9 + 7 + 6

Perimeter of ΔABC = 22 cm

Semi perimeter of triangle ΔABC

Semi Perimeter of ΔABC

Semi Perimeter of ΔABC = 11 cm

Area of ΔABC

Area of ΔABC

Area of ΔABC = 2√110 cm²

Area od Quadrilateral ABCD = Area of ΔACD + Area of ΔABC

Area od Quadrilateral ABCD = 54 + 2√110

Area of Quadrilateral ABCD = (54 + 2√110) cm²