Question

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

Options

• 2 × area of ΔABD

• 2 × area of ΔBCD

• 4 × area of ΔAOB

• 2 × area of ΔABC

Answer 1

ABCD is a quadrilateral whose diagonals intersect each other at point O. The diagonal AC bisects diagonal BD. Then area of quadrilateral ABCD is 2 × area of ΔABD.

Explanation:

• The diagonal AC divides the quadrilateral ABCD into two triangles: ΔABD and ΔCBD.
• Since diagonal AC bisects diagonal BD at point O, point O is the mid-point of BD.
• Therefore, segment DO = OB, which means AO acts as a median for both ΔABD (dividing it into ΔAOD and ΔAOB of equal area) and ΔCBD (dividing it into ΔCOD and ΔCOB of equal area).
• Because the area of ΔABD is equal to the area of ΔCBD (both sharing base BD and equal heights DO and OB from vertex A and C respectively, or simply being split symmetrically by the bisecting diagonal), the total area of the quadrilateral is the sum of these two equal triangle areas:
  area of quadrilateral ABCD = area of ΔABD + area of ΔCBD = 2 × area of ΔABD