SOLUTION: show that the quadrilateral formed by joining the mid-points of the sides of a square is also a square.

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Question 161720: show that the quadrilateral formed by joining the mid-points of the sides of a square is also a square.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Given:
P, Q, R and S+are themid-points+of AB, BC, CD and DA respectively.
To prove:
PQRS is a rhombus.
Proof:
Consider triangle BAC+
PQ|| AC and PQ+=+%281%2F2%29AC….(1)
(In a triangle the segment joining the mid-points of
two sides are parallel and equal to third side)
Consider triangle ADC+,
SR|| AC and SR+=+%281%2F2%29AC….(2)


From (1) and (2),
PQ|| SR and PQ=SR
PQRS is a parallelogram ……………….(3)
AD+=+BC….. (opposite sides of a rectange)
So
%281%2F2%29AD+=%281%2F2%29BC…..
i.e. AS+=+BQ+
Consider triangle +APS+, and triangle +BPQ+,
AP+=+BP…. (P is the mid-point of AB)
AS+=+BQ
angle SAP = PBQ = 90degrees
So, triangle +APS+ is congruent to triangle +BPQ+.... (SAS congruency condition)

+PS+=+PQ ………………………..(4)
From (3) and (4),
PQRS is a parallelogram in which PS+=+PQ+.
PQRS is a rhombus .