ABCD is a trapezium ,AB is parallel to DC .P and Q are the mud points of AC and BD.Prove that PQ//AB//DC and PQ =1/2(AB-DC) Share with your friends Share 0 Lovina Kansal answered this Dear student Given: A trapezium ABCD in which AB∥DC and P and Q are the mid-pointsof its diagonals AC and BD respectively.To prove: (i) PQ∥AB or DC (ii) PQ=12AB-DCConstruction: Join DP and produce DP to meet AB in R.Proof: Since AB∥DC and transversal AC cuts them at A and C respectively.∴∠1=∠2 ...(1) [∴Alternate angles are equal]Now, In △s APR and DPC, we have∠1=∠2 [from 1]AP=CP ∵P is the mid-point of ACand ∠3=∠4 [Vertcally opposite angles]So, by ASA criterion of congruence, we have△APR≅△DPC⇒AR=DC and PR=DP [C.P.C.T] ...(2)In △DRB,P and Q are the mid-points of sides DR and DB respectively.∴PQ∥RB⇒PA∥AB [∵RB is a a part of AB]PQ∥AB and DC [∵AB∥DC (Given)]Again,P and Q are the mid-points of sides DR and DB respectively in △DRB.∴PQ=12RB⇒PQ=12AB-AR⇒PQ=12AB-DC ∵from (2), AR=DC Regards 2 View Full Answer Ria Tosh answered this BLAH 0