Question

In a given PQRS is a parallelogram in which PQ is produced to T such that QT=PQ Prove that ST bisects RQ

Answer 1

since q is the midpoint and qo || sp (rq || sp since parallelogram , oq || sp )
oq =1/2 sp (mid point theorem)
2oq =sp __ (1)
but sp = rq (opposite sides of a parallelogram) _______ (2)
from (1) and (2) ,
2oq =sp
oq = 1/2 sp
o is the midpoint of rq
therefore , St bisects rq


hope it helps !

Answer 2

Proved below.

Step-by-step explanation:

Given:

We have PQRS is a parallelogram.

As shown in the figure below,

PQ = SR  [opposite sides of parallelogram are equal]

But PQ = QT  [given]  

So, SR = QT                    [1]  

Since SR || PT and ST is a transversal

∠RSM = ∠QTM  [Alternate interior angles]

In triangle SMR and TMQ

,∠SMR = ∠TMQ  [alternate interior angles]

∠RSM = ∠QTM  [Alternate interior angles]

SR = QT  [from 1]

ΔSMR is congruent to ΔTMQ  [AAS CRITERIA]

⇒ RM = QM  [CPCT]

So, ST bisects RQ.

Hence proved.