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Question
ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the mid-points of AB, AC, CD and BD respectively, show that PQRS is a rhombus.
Solution
AD = BC and P, Q, R and S are the mid-points of sides AB, AC, CD and BD respectively, show that PQRS is a rhombus.
In ΔBAD, by mid-point theorem
PS || AD and PS `=1/2` AD .......(i)
In ΔCAD, by mid-point theorem
QR || AD and QR `=1/2` AD .......(ii)
Compare (i) and (ii)
PS || QR and PS = QR
Since one pair of opposite sides is equal as well as parallel then
PQRS is a parallelogram ...(iii)
Now, In ΔABC, by mid-point theorem
PQ || BC and PQ `=1/2` BC .......(iv)
And, AD = BC …(v) [given]
Compare equations (i) (iv) and (v)
PS = PQ …(vi)
From (iii) and (vi)
Since, PQRS is a parallelogram with PS = PQ then PQRS is a rhombus.
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