The figure formed by joining the mid-points of the adjacent sides of a square is a
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,
In quadrilateral EHOS, we have
∠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.