ABCD is a parallelogram. AC and BD are the diagonals intersect at O. P and Q are the points of tri section of the diagonal BD. Prove that `CQ" ||" AP` and also AC bisects PQ (see figure).

Let A B C D be a p[arallelogram whose diagonals intersect at P and let O be the origin. Then prove that vec O A+ vec O B+ vec O C+ vec O D=4 vec O Pdot

ABCD is a rectangle whose diagonals AC and BD intersect at O. If angleOAB= 46^(@) , find angleOBC .

The diagonal AC of a parallelogram ABCD intersects DP at the point Q, where ‘P’ is any point on side AB. Prove that CQ xx PQ = QA xx QD .

ABCD is a parallelogram and AP and CQ are perpendiculars from vertex A and C on diagonal BD. Show that (i) DeltaAPB~=DeltaCQD (ii) AP=CQ .

ABCD is a quadrilateral in which AB= AD, the bisector of angle BAC and angle CAD intersect the sides BC and CD at the points E and F respectively. Prove that EF || BD.

ABCD is a parallelogram. L is a point on BC which divides BC in the ratio 1:2 . AL intersects BD at P.M is a point on DC which divides DC in the ratio 1 : 2 and AM intersects BD in Q. Point Q divides DB in the ratio

ABCD is a parallelogram with A as the origin. vec b and vec d are the position vectors of B and D respectively. a.What is the position vector of C? (b) What is the angle between vec(AB) and vec (AD) ? (c) Find vec(AC) (d) If |vec(AC)| = |vec (BD)| , show that ABCD is rectangle.

Let O A C B be a parallelogram with O at the origin and O C a diagonal. Let D be the midpoint of O Adot using vector methods prove that B Da n dC O intersect in the same ratio. Determine this ratio.