Diagonals of a Rectangle Bisect Each-Other and Are Equal
ABCD is a qua...
Question
ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA. AC is the diagonal. Show that
(i) SR || AC and SR=12AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.
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Solution
(i) In △ADC, R is the mid-point of DC and S is the mid-point of DA.
Thus, by mid-point theorem, SR||AC and SR=12AC.
(ii) In △BAC, P is the mid-point of AB and Q is the mid-point of BC.
Thus, by mid-point theorem, PQ || AC and PQ=12AC.
Also, SR=12AC
Hence, PQ = SR.
(iii) SR || AC ... From Question (i)
PQ || AC ... From Question (ii) ⇒PQ||SR
From (ii), PQ = SR
Since, one pair of opposides of the quadrilateral PQRS is parallel and equal, PQRS is a Parallelogram.
Hence Proved.