Show that quadrilateral obtained by joining the midpoints of a square is also a square
Answers
Step-by-step explanation:
Given :-
A square
Required to prove :-
The quadrilateral obtained by joining the midpoints of a square is also a square.
Proof :-
Let consider ABCD is a square.
We know that
All sides are equal in a square
Therefore, AB = BC = CD = DA
The mid point of AB = P
AP = PB
Therefore, AB = 1/2 AP = 1/2 PB
The mid point of BC = Q
Therefore, BQ = QC
Therefore, BC = 1/2 BQ = 1/2 QC
The mid point of CD = R
CR = RD
Therefore, CD = 1/2 CR = 1/2 RD
The mid point of DA = S
DS = SA
Therefore, DA = 1/2 DS = 1/2 SA
Now,
In ∆ DAC,
S is the mid point of DA
R is the mid point of CD
Therefore,
SR = 1/2 AC ------------(1)
And
PQ = 1/2 AC ------------(2)
PS = 1/2 BD = 1/2 AC -------(3)
We know that
Diagonals are equal in a square
=> BD = AC ------------(4)
=> 1/2 BD = 1/2 AC
Therefore, QR = 1/2 BD = 1/2 AC -----(5)
From all the above equations
PQ = QR = RS = PS
and
The diagonals = PR and QS
PR = BC
QS = AB
QS = PR
AB = BC
Therefore, All sides are equal in the square PQRS .
and Diagonals are equal.
Therefore, PQRS is a square.
Hence, Proved.
