Question

In the adjoining figure ABCD is a parallelogram in which p is the midpoint of DC and q is a point on ac such that CQ =1/4 AC Also PQ when produced meet BC at R Prove that R is the midpoint of BC​

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Asked on December 26, 2019 by

Shivu Jawaid

A parallelogram ABCD has P the mid- point of DC and Q intersects AC such that CQ=

4

1

AC. PQ produced meets BC at R prove that :

(a) R is the mid-point of BC

(b) PR=

2

1

DB

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ANSWER

Given ABCD is a parallelogram and P is midpoint of DC

also, CQ=

4

1

AC

To Prove : R is mid point of BC

Proof : Now

OC=

2

1

AC (Diagonals of parallelogram bisect each other) ...(i)

and CD=

4

1

AC ...(ii)

From (i) and (ii)

CD=

2

1

OC

In ΔDCO P and Q are midpoint of DC and OC Respectively

∴PQ∥DO

Also in ΔCOB Q is midpoint of OC and PQ∥DB

∴R is midpoint of BC

∴ in ΔABCPR∥DB

CD

CP

=

CB

CR

=

BD

PR

DB

PR

=

2

1

∴PR=

2

1

DB