Questions on Geometry: Proofs in Geometry answered by real tutors!

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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.: show that the quadrilateral formed by joining the mid-points of the sides of a square is also a square.
Answer by MathLover1(1160) 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 = (1/2)AC….(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 = (1/2)AC….(2)


From (1) and (2),
PQ|| SR and PQ=SR
PQRS is a parallelogram ……………….(3)
AD = BC….. (opposite sides of a rectange)
So
(1/2)AD =(1/2)BC…..
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 .