Advertisements
Advertisements
प्रश्न
In the figure, ABCD is a parallelogram and CP is parallel to DB. Prove that: Area of OBPC = `(3)/(4)"area of ABCD"`
Advertisements
उत्तर
Since the diagonals of a parallelogram divide it into four triangles of equal area
Therefore, area of ΔAOD = area ΔBOC = area ΔABO = area ΔCDO.
⇒ area ΔBOC = `(1)/(4)`area(||gm ABCD) ..........(i)
In ||gm ABCD, BD is the diagonal
Therefore, area(ΔABD) = area(ΔBCD)
⇒ area (ΔBCD) = `(1)/(2)`area(||gm ABCD) .......(ii)
In ||gm BPCD, BC is the diagonal
Therefore, area(ΔBCD) = area(ΔBPC) .....(iii)
From (iii) and (ii)
⇒ area (ΔBPC) = `(1)/(2)`area(||gm ABCD) .......(iv)
adding (i) and (iv)
area(ΔBPC) + areaΔBOC = `(1)/(2)"area(||gm ABCD)" + (1)/(4)"area(||gm ABCD)"`
Area of OBPC = `(3)/(4)"area of ABCD"`.
APPEARS IN
संबंधित प्रश्न
ABCD is a parallelogram. P and Q are mid-points of AB and CD. Prove that APCQ is also a parallelogram.
ABCD is a parallelogram. P and T are points on AB and DC respectively and AP = CT. Prove that PT and BD bisect each other.
PQRS is a parallelogram. PQ is produced to T so that PQ = QT. Prove that PQ = QT. Prove that ST bisects QR.
Prove that if the diagonals of a parallelogram are equal then it is a rectangle.
P is a point on side KN of a parallelogram KLMN such that KP : PN is 1 : 2. Q is a point on side LM such that LQ : MQ is 2 : 1. Prove that KQMP is a parallelogram.
In a parallelogram PQRS, M and N are the midpoints of the opposite sides PQ and RS respectively. Prove that
PMRN is a parallelogram.
In a parallelogram PQRS, M and N are the midpoints of the opposite sides PQ and RS respectively. Prove that
MN bisects QS.
In the given figure, PQRS is a trapezium in which PQ ‖ SR and PS = QR. Prove that: ∠PSR = ∠QRS and ∠SPQ = ∠RQP
Prove that the diagonals of a kite intersect each other at right angles.
The diagonals AC and BC of a quadrilateral ABCD intersect at O. Prove that if BO = OD, then areas of ΔABC an ΔADC area equal.
