pqrs is a trapezium in which pq is parallel to rs a,b,c,d are the mid points of pq - Maths - Areas of Parallelograms and Triangles

Question

pqrs is a trapezium in which pq is parallel to rs a,b,c,d are the
mid points of pq , qs , rs , pr respectively, show that abcd is a
rhombus

Anuradha Sharma · Accepted Answer

Given- [Let ABCD be a given isosceles trapezium where AB and DC are
the parallel sides and AD and BC are the oblique sides(
non-parallel sides) which are equal P, Q, R and S are the mid
points of sides AB, BC, CD and DA respectively P, Q; Q, R; R, S ans
S,P are joined The quadrilateral PQRS is formed ]

Required To Prove- [PQRS is a rhombus]

Construction- [The diagonals AC and BD of ABCD are joined
]

Proof- In triangle DCB,

R and Q are the mid points of sides DC and BC
respectively

Therefore, RQ=1/2 BD and RQ parallel to
BD------------->1

In triangle DAB,

S and P are the mid points of sides AD and AB
respectively

Therefore, SP=1/2 BD and SP parallel to
BD----------->2

In quadrilateral PQRS,

RQ=SP and RQ parallel to SP [ FROM 1 AND 2 ]

Therefore, PQRS is a parallelogram

In triangle ACB,

P and Q are the mid points of sides AB and BC
respectively

Therefore, PQ=1/2 AC and PQ parallel to
AC--------->3

In triangle DAC,

S and R are the mid points of sides AD and DC
respectively

Therefore, SR=1/2 AC and SR parallel to
AC----------->4

Now, the diagonals of an isosceles trapezium are equal
AC=BD

So, from 1, 2, 3 and 4 we have,

1/2 AC=1/2 BD=PQ=QR=RS=SP

In parallelogram PQRS,

PQ=QR=RS=SP

Therefore, PQRS is a rhombus

[Proved]

pqrs is a trapezium in which pq is parallel to rs. a,b,c,d are the mid points of pq , qs , rs , pr respectively, show that abcd is a rhombus