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  • 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.

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.

Answers (1)

Answer:  [D]

Solution.
According to question, the quadrilateral ABCD is formed by joining the midpoints of PQRS
If PQRS is a rhombus

Here ABCD is not a rhombus because in rhombus angles need not be right angles.
But here \angle A=\angle B=\angle C=\angle D=90^{\circ}
ABCD is a square not a rhombus.
If PQRS is a parallelogram

Here ABCD is not a rhombus because in rhombus all sides of if will be equal here sides of quadrilateral ABCD are not equal
it is not a rhombus
If diagonals of PQRS are perpendicular

Here AB D is not a rhombus because in rhombus angles are not right angles and all sides are equal but here AB = CD and BC = AD also \angle A=\angle C=\angle D=90^{\circ}
Hence ABCD is a rectangle not a square.
If diagonals of PQRS are equal

In ABCD quadrilateral here all sides are equal and angles of quadrilateral ABCD is not right angle, therefore, ABCD is a rhombus\

\ option D is correct
(D) diagonals of PQRS are equal

 

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