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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)

According to the 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 a 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 a rhombus all sides of it 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 a 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 the angles of quadrilateral ABCD are not right angles, therefore, ABCD is a rhombus\

\ Option D is correct
(D) diagonals of PQRS are equal.

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