Question

ABCD is a parallelogram. if P and Q are points on AD and BC respectively. such that AP equals to 1\3 AD and CQ is equals to 1\3 BC, prove that AQCP is a parallelogram.

Answer 1

Given:

Parallelogram = ABCD

Side = AD and BC with mid points P and Q

AP = 1/3AD

CQ= 1/3BC

To Find:

AQCP is a parallelogram

Solution:

In △ABQ and △CDP

AB=CD

Thus, ∠B=∠D

Now,

DP = AD−PA

DP = 2/3QD

Similarly.

BQ = BC−CQ

BQ = BC - 1/3

Since, AD=BC

BQ =  2/3BC = 2/3AD

= BQ=DP

By SAS congruence criterion

△ABQ≅△CDP

AQ=CP (c.p.c.t)

PA=  1/3PD

CQ = 1/3BC = 1/3AD 

Hence,

​PA=CQ

∠QAB=∠PCD (c.p.c.t) --- eq 1

∠QAP =∠A−∠QAB

Taking equation 1 -

∠A=∠C

∠Q=∠C−∠PCD

As the alternate interior angles are equal, thus -

∠QAP=∠PCQ

Therefore, AQ and CP are two parallel lines

Answer: Therefore it is proved that PAQC is a parallelogram