Find the area of a quadrilateral one of whose diagonals is 30 cm long and the perpendiculars from the other two vertices are 19 cm and 11 cm respectively.

The quadrilateral is shown in the figure below:

Area of triangle = $\dfrac{1}{2}$ x base x height

For Δ ABD,

Area = $\dfrac{1}{2}$ x BD x AM

= $\dfrac{1}{2}$ x 30 x 19 cm²

= 15 x 19 cm²

= 285 cm²

For Δ CBD,

Area = $\dfrac{1}{2}$ x BD x CN

= $\dfrac{1}{2}$ x 30 x 11 cm²

= 15 x 11 cm²

= 165 cm²

Area of quadrilateral ABCD = Area of Δ ABD + Area of Δ CBD

= 285 + 165 cm²

= 450 cm²

Hence, the area of quadrilateral is 450 cm².