Question

proove tat the quadrilateral formed by joining the mid points of consecrive sides of a square, is also a square
​

Answer 1

In a square 

ABCD,

 

P,Q,R

 and 

S

 are the mid-points of 

AB,BC,CD

 and 

DA

 respectively.

⇒

 

  

AB=

BC=

CD=

AD

           [ Sides of square are equal ]

In 

△ADC,

SR=

SR∥AC

2

1

AC

In 

△ABC,

PQ=

PQ∥AC

2

1

AC

From equation ( 1 ) and ( 2 ),

SR=

PQ=

SR∥PQ

2

1

AC

Similarly, 

SP∥

BD

and 

BD∥

RQ

∴

 

  

SP∥

RQ

and 

SP=

2

1

BD

and 

RQ=

2

1

BD

∴

 

  

SP=

RQ=

2

1

BD

Since, diagonals of a square bisect each other at right angle.

∴

 

  

AC=

BD

⇒

 

  

SP=

RQ=

2

1

AC

          ----- ( 4 )

From ( 3 ) and ( 4 )

SR=PQ=SP=RQ

We know that the diagonals of a square bisect each other at right angles.

∠EOF=90

o

.

Now, 

RQ∥

DB

RE∥FO

Also, 

SR∥

AC

⇒  

FR∥

OE

∴OERFis a parallelogram.

So, 

∠FRE=

∠EOF=

90

o

  (Opposite angles are equal)

Thus,

PQRS

is a parallelogram with 

∠R=

90

o

and 

SR=

PQ=

SP=

RQ

∴PQRSis a square.