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If diagonals of a parallelogram ABCD intersect each other at point O, then find the ratio of area of ΔAOB and area of ABCD.

If diagonals of a parallelogram ABCD intersect each other at point O, then find the ratio of area of ΔAOB and area of ABCD. Correct Answer 1 : 4

Given:

ABCD is a parallelogram

Concept used:

Diagonals of parallelogram bisect each other.

Diagonal of a parallelogram divides the parallelogram in two triangles of equal area.

Median of a triangle divides the triangle into two parts having equal area.

Calculation:

Area of ΔADC = Area of ΔABC (As, Diagonal of a parallelogram divides the parallelogram in two triangles of equal area)

AO = OC and DO = OB (As, Diagonals of parallelogram bisect each other)

In ΔABC, AO = OC

⇒ BO is a median

⇒ Area of ΔAOB = Area of ΔBOC (Median of a triangle divides the triangle into two parts having equal area)

Similarly, Area of ΔAOD = Area of ΔDOC

And, Area of ΔAOD = Area of ΔAOB

⇒ Area of ΔAOB = Area of ΔBOC = Area of ΔAOD = Area of ΔDOC

Area of parallelogram = Area of ΔAOB + Area of ΔBOC + Area of ΔAOD + Area of ΔDOC

⇒ Area of parallelogram = 4 × Area of ΔAOB

⇒ Area of ΔAOB/Area of parallelogram = 1/4

∴ Area of ΔAOB : Area of parallelogram is 1 : 4

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