P is the mid-point of the side AB of a parallelogram ABCD. A line thro

1.	P is the mid-point of the side AB of a parallelogram ABCD. A line through B parallel toPD meets DC at Q and AD produced at R. Prove that AR = 2BC and BR = 2BQ.
Answer» In triangle ABR, P is the mid point of AB and DP is parallel to BR (by question).Thus D is the mid point of AR. So we have AR = 2 AD = 2 BC ( since in parallelogram ABCD opp. sides AD = BC) (1) Again in triangle ABR, since BR is parallel to DP (or, since Trangles ABR and ADP are similar) we have BR /DP = AB/DP = 2 ( since P is the mid pt. of AB) Therefore BR = 2 DP = 2 BQ ( since PBQR is a parallelogram. Fot DQ is parralel to AB and BQ is drawn parallel to PD)

P is the mid-point of the side AB of a parallelogram ABCD. A line through B parallel toPD meets DC at Q and AD produced at R. Prove that AR = 2BC and BR = 2BQ.

Answer»

In triangle ABR, P is the mid point of AB and DP is parallel to BR (by question).Thus D is the mid point of AR.

So we have AR = 2 AD = 2 BC ( since in parallelogram ABCD opp. sides AD = BC) (1)

Again in triangle ABR, since BR is parallel to DP (or, since Trangles ABR and ADP are similar)

we have BR /DP = AB/DP = 2 ( since P is the mid pt. of AB)

Therefore BR = 2 DP = 2 BQ ( since PBQR is a parallelogram. Fot DQ is parralel to AB and BQ is drawn parallel to PD)