ABCD is a quadrilateral whose diagonals intersect each other at point O. The diagonal AC bisects diagonal BD. Then area of quadrilateral ABCD is :

1. 2 x area of ΔABD

2. 2 x area of ΔBCD

3. 4 x area of ΔAOB

4. 2 x area of ΔABC

ABCD is a quadrilateral. Diagonals AC and BD intersect at point O. Diagonal AC bisects diagonal BD.

Since AC bisects BD at O. O is the midpoint of BD. 

Using the property, a median of a triangle divides it into two triangles of equal area.

⇒ AO is a median of ΔABD and CO is a median of ΔCBD. 

⇒ Area of (ΔAOD) = Area of (ΔAOB) as AO is a median to BD in ΔABD.

⇒ Area of (ΔCOD) = Area of (ΔCOB) as CO is a median to BD in ΔCBD.

As we know that Area of quadrilateral ABCD = Area of (ΔABD) + Area of (ΔBCD)

⇒ Area of quadrilateral ABCD = [Area of (ΔAOD) + Area of (ΔAOB)] + [Area of (ΔCOD) + Area of (ΔCOB)]

⇒ Area of quadrilateral ABCD = [Area of (ΔAOB) + Area of (ΔAOB)] + [Area of (ΔCOB) + Area of (ΔCOB)]

⇒ Area of quadrilateral ABCD = 2Area of (ΔAOB) + 2Area of (ΔCOB)

⇒ Area of quadrilateral ABCD = 2[Area of (ΔAOB) + Area of (ΔCOB)]

⇒ Area of quadrilateral ABCD = 2 x Area of (ΔABC).

Hence, option 4 is the correct option.