a In figure 1 given below ABCD is a parallelogram and X is the midpoint of BC The line AX produced m...

Question

a In figure 1 given below ABCD is a parallelogram and X is the midpoint of BC The line AX produced meets DC produced at Q The parallelogram ABPQ is completed Prove that i the triangles ABX and QCX are congruent iiDC CQ QP b In figure 2 given below points P and Q have been taken on opposite sides AB and CD respectively of a parallelogram ABCD such that AP CQ Show that AC and PQ bisect each other

Accepted Answer

a Given ABCD is parallelogram and X is midpoint of BC The line AX produced meets DC produced at Q and ABPQ is a gm To prove i ABX QCX ii DC CQ QP Proof In ABX and QCX we have BX XC X is the midpoint of BC AXB CXQ Vertically opposite angles XCQ XBA Alternate angle since AB CQ So ABX QCX by ASA axiom of congruence Now by CPCT CQ AB But AB DC and AB QP As ABCD and ABPQ are gms Hence DC CQ QP b In gm ABCD P and Q are points on AB and CD respectively PQ and AC intersect each other at O and AP CQ To prove AC and PQ bisect each other ie AO OC and PO OQ Proof In AOP and COQ AP CQ Given AOP COQ Vertically opposite angles OAP OCP Alternate angles So AOP COQ by AAS axiom of congruence Now by CPCT OP OQ and OA OC Hence proved

ML Aggarwal Solutions Class 9 Mathematics Solutions for Rectilinear Figures Exercise 13.1 in Chapter 13 - Rectilinear Figures