In the figure (2) given below, PQRS is a parallelogram formed by drawing lines parallel to the diagonals of a quadrilateral ABCD through its corners. Prove that area of || gm PQRS = 2 × area of quad. ABCD.

From figure,

Area of ∆ACD = $\dfrac{1}{2}$ Area of || gm ACRS [As both are on same base AC and between the same parallel lines AC and SR]

⇒ Area of || gm ACRS = 2 Area of ∆ACD …….(i)

Similarly,

Area of ∆ABC = $\dfrac{1}{2}$ Area of || gm ∆APQC [As both are on same base AC and between the same parallel lines AC and PQ]

⇒ Area of || gm APQC = 2 Area of ∆ABC …….(ii)

Adding (i) and (ii),

⇒ Area of || gm ACRS + Area of || gm APQC = 2 Area of ∆ACD + 2 Area of ∆ABC

⇒ Area of || gm PQRS = 2[Area of ∆ACD + Area of ∆ ABC]

⇒ Area of || gm PQRS = 2(Area of quad. ABCD)

Hence, proved that area of || gm PQRS = 2 x area of quad. ABCD.