PQRS is a parallelogram. A is midpoint of PQ and C is a point on diagonal SQ such that
QC= 1
4
QS. AC produced meets QR at B. Prove that: “B” is midpoint of QR and AB is half of PR.
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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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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
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