Question

30. ABCD is a quadrilateral in which AD = BC. If P, Q, R and S are midpoints of AB, AC, DC and
DB prove that PQRS is a rhombus.
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Answer 1

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Answers:-

Here is your answer:

First of all we will draw a quadrilateral ABCD with AD = BC and join AC, BD, P,Q,R,S are the mid points of AB, AC, CD and BD respectively.

In the triangle ABC, P and Q are mid points of AB and AC respectively.

PQ || BC and PQ = 1/2

BC .. (1)

In ΔADC, QR = 1/2

AD = 1/2

BC ... (2)

Now we will consider ΔBCD,

SR = 1/2

BC.. (3)

In ΔABD,

PS = 1/2

AD = 1/2

BC.. (4)

So from (1), (2), (3) and (4)

we will get

PQ = QR = SR = PS

All sides are equal so PQRS is a Rhombus.

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Answer 2

Answer:

Given :-

A quadrilateral ABCD

AD=BC

AB=CD

P,Q,R,S are the mid points

To PROVE:−

PQRS IS RHOMBUS

So, PS=PQ=QR=RS

In ∆ BCD,

By mid-pint theorem,

Q is the midpoint of BC and R is the midpoint of DC.

BD∣∣QR,QR= 1/2BD

QR=1/2 BD .....(1)

In ∆ BAD,

By mid-pint theorem,

P is the midpoint of AB and S is the midpoint of AD.

So,

SP=1/2 BD .....(2)

By (1) and (2),

Therefore, □ PQRS is parallelogram.

Now, In ∆ ABC,

By mid-pint theorem,

P is the midpoint of AB and Q is the midpoint of BC.

PQ∣∣AC,PQ=1/2AC

PQ=1/2 AC .......(3)

Now, we know that AC=BD,

Therefore, (3) becomes,

PQ∣∣BD,BD=1/2PQ

BD=1/2PQ .......(4)

Therefore,PS=PQ=QR=RS

As, all sides are equal it is a rhombus.

Hence,proved!!!