Question

ABCD is a rhombus with P,Q,R as mid points of AB,BC,CD respectively. Prove that PQ is perpendicular to QR

Answer 1

Given: ABCD is a rhombus
           P, Q, R  are then mid points of  AB, BC, CD.
to proove: angle PQR = 90
Construction: Join  AC and BD.
Proof :   In ΔDBC

R and Q are the mid points of DC and CB
.'.RQ // DB                                   ( by mid point theorem)
.'. MQ//ON         ..............eq.1      (parts of RQ and DB)

Now , in Δ ACB
P and Q are the mid points of  AB and BC
.'. AC // PQ                                  ( by mid point theorem)
.'. OM //NQ        .................eq.2     ( OM and NQ are the parts of AC and PQ)

from eq. 1 and 2

MQ //ON
ON //NQ
'.' each pair of opposite side is parallel

.'.ONQM is a parallelogram

in ONQM
angle MON = 90'          (because diagonals of rhombus bisect each other at 90')

angle MON = angle PQR       ( opposite angle of parallelogram)
.'. angle PQR = 90'

Answer 2

Refer to the solution given.

hope it helps.