Find the area of a quadrilateral one of whose diagonals is 25 cm and the lengths of perpendiculars from the other two vertices are 16.4 cm and 11.6 cm respectively.

A quadrilateral ABCD is shown in the figure below :

AM and CN are the perpendiculars from A and C respectively to the diagonal BD.

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

For △ABD,

Area = $\dfrac{1}{2}$ × BD × AM

= $\dfrac{1}{2}$ × 25 × 16.4

= 25 × 8.2

= 205 cm².

For △CBD,

Area = $\dfrac{1}{2}$ × BD × CN

= $\dfrac{1}{2}$ × 25 × 11.6

= 25 × 5.8

= 145 cm².

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

= 205 + 145

= 350 cm².

Hence, area of quadrilateral = 350 cm².