BCD is a parallelogram. Point P divides AB in theratio 2:3 and point Q divides DC in the ratio 4:1.Prove that OC is half of OA.2A BCDPQO

ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig 8.29). AC is a diagonal. Show that:(i) SR || AC and SR = 1/2 AC(ii) PQ = SR(iii) PQRS is a parallelogram.

OABC is a square and the line segments OA, OB, OC are represented by the vectors a, b and c respectively. If D is the midpoint of BC, show that the line segment OD is represented by either c + 1/2a or b - 1/2a or 1/2(b + c). Verify that each of these can be obtained from either of the others by using the relation b = a + c

P is the mid-point of the side CD of a parallelogram ABCD. A line through C parallel to PA intersects AB at Q and DA produced at R. Prove that DA=AR and CW

If OC + OA = 2OB, prove that A, B, and C are colinear and B is the midpoint of AC.

In parallelogram ABCD, two point P and Q are taken on diagonal BD such that DP=BQ. Show that(i) △APD≅△CQB(ii) AP=CQ(iii) △AQB≅△CPD(iv) AQ=CP(v) APCQ is a parallelogram