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# ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

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#### Here, we are joining A and C.In ΔABCP is the mid point of ABQ is the mid point of BCPQ||AC [Line segments joining the mid points of two sides of a triangle is parallel to AC(third side) and also is half of it]PQ=12ACIn ΔADCR is mid point of CDS is mid point of ADRS||AC [Line segments joining the mid points of two sides of a triangle is parallel to third side and also is half of it]RS=12ACPQ||RS and PQ=RS So, PQRS is a parallelogram. [one pair of opposite side is parallel and equal]In ΔAPS & ΔBPQAP=BP [P is the mid point of AB)∠PAS=∠PBQ(All the angles of rectangle are 90o)AS=BQ∴ΔAPS≅ΔBPQ(SAS congruency)∴PS=PQPS=RQ & PQ=RS (opposite sides of parallelogram is equal)PQ=RS=PS=RQ[All sides are equal]PQRS is a parallelogram with all sides equal∴ So PQRS is a rhombus.

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