Question

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rhombus if

• A. PQRS is a rhombus
• B. PQRS is a parallelogram
• C. diagonals of PQRS are perpendicular
• D. diagonals of PQRS are equal

Answer 1

Answer: (A)

PQRS is a rhombus

Let us join AC and BD

In triangle ABC

P and Q are the mid point of AB ,AC respt

So PQ II AC and $PQ=\frac{1}{2}AC..........\left(1\right)$

Similarly triangle ADC

SR II AC and $SR=\frac{1}{2}AC..........\left(2\right)$

Clearly PQ II SR and PQ = SR

Since in quadrilateral PQRS one pair of side is equal and parallel

to each parallelogram

So PQ II QR and PS=PR ........................(3)

In triangle BCD Q and R are mid point of side BC and CD respt

So bQR II BD and QR=$Qr=\frac{1}{2}BD..........\left(4\right)$

However the diagonal of rectangle is equal

By using eq (1) ,(2),(3) and (4)

PQRS is a rhombus