Question

ABCD is a parallelogram and Of is any point on its diagonal AC. Show that are(AOB)=ar(AOD)

Answer 1

Join BD intersecting AC at M,

In ∆DMO and ∆BMO , OM is the median
•°• ar(DMO) = ar(BMO)

ar(ADC) = ar(ABC) ( since diagonal of a ||gm divides it into two triangles of equal area)

(also Diagonals of a parallelogram divides it into four triangles of equal area)

•°• ar(AMD) = ar(AMB)
ar(AMD) + ar(DMO) = ar(AMD) + ar(DMO) ( adding ar(DMO) both sides)
Thus, ar(AOD) = ar(AOB)

Answer 2

hence area of triangle aob is equal to area of triangle aod