# The figure formed

Let ABCD is a square such that AB = BC = CD = DA, AC = BD and P, Q, R and S are the mid points of the sides AB, BC, CD and DA respectively.

In

P and Q are the mid-points of AB and BC respectively.

Therefore,

PQ AC and PQ = AC (Mid-point theorem) (i)

Similarly,

In

SR AC and SR = AC (Mid-point theorem) (ii)

Clearly,

PQ SR and PQ = SR

Since, in quadrilateral PQRS, one pair of opposite sides is equal and parallel to each other. Hence, it is a parallelogram.

Therefore,

PS QR and PS = QR (Opposite sides of a parallelogram) (iii)

In

Q and R are the mid-points of sides BC and CD respectively

Therefore,

QR BD and QR = BD (Mid-point theorem) (iv)

However, the diagonals of a square are equal

Therefore,

AC = BD (v)

By using equation (i), (ii), (iii), (iv) and (v), we obtain

PQ = QR = SR = PS

We know that, diagonals of a square are perpendicular bisector of each other

Therefore,

AOD = AOB = COD = BOC = 90o

Now, in quadrilateral EHOS, we have

SE || OH

Therefore,

AOD + AES = 180o (Corresponding angle)

AES = 180° - 90°

= 90°

Again,

AES + SEO = 180o (Linear pair)

SEO = 180° - 90°

= 90°

Similarly,

SH || EO

Therefore,

AOD + DHS = 180o (Corresponding angle)

DHS = 180° - 90° = 90°

Again,

DHS + SHO = 180° (Linear pair)

SHO = 180° - 90°

= 90°

Again,

SEO = SHO = EOH = 90°

Therefore, by angle sum property of quadrilateral in EHOS, we get

SEO + SHO + EOH + ESH = 360°

90o + 90o + 90o + ESH = 360°

ESH = 90°

In the same manner, in quadrilateral EFOP, FGOQ, GHOR, we get

HRG = FQG = EPF = 90°

Therefore, in quadrilateral PQRS, we have

PQ = QR = SR = PS and ESH = HRG = FQG = EPF = 90°

Hence, PQRS is a square.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Conservation of Energy41 mins
Newton's Second Law46 mins
Division of Polynomials46 mins
Pair of Angles29 mins
Newton's Third Law44 mins
Miscellaneous questions42 mins
Newton's First Law45 mins
Significance of Newton's Laws in daily life42 mins
Equations of Motion41 mins
Final Phase & Legacy of the Revolution43 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses