Question -
Answer -
Given: Here from the given figure we get
(1) ABCD is a parallelogram
(2) BD and CA are the diagonals intersecting at O.
(3) P is any point on BO
To prove:
(a) Area of ╬ФADO = Area of╬Ф CDO
(b) Area of ╬ФAPB = Area of╬Ф CBP
Proof: We know that diagonals of a parallelogram bisect each other.
┬аO is the midpoint of AC and BD.
Since medians divide the triangle into two equal areas
In ╬ФACD, DO is the median
┬аArea of ╬ФADO = Area of╬Ф CDO
Again O is the midpoint of AC.
In ╬ФAPC, OP is the median
тЗТ┬аAreaof┬а├ДAOP┬а=Area of┬а├ДCOP┬атАжтАж (1)
Similarly O is the midpoint of AC.
In ╬ФABC, OB is the median
тЗТ┬аAreaof┬а├ДAOB┬а=Area of┬а├ДCOB┬атАжтАж (2)
Subtracting (1) from (2) we get,
Area of ╬ФAOB тИТ Area of ╬ФAOP = Area of ╬ФCOB тИТ Area of ╬ФCOP
тЗТ┬аAreaof┬а├ДABP┬а=Area of┬а├ДCBP
Hence it is proved that
(a)┬а
(b)┬а