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The quadrilateral formed by joining the mid – points of the side of 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.

User Zira
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1 Answer

3 votes

Final answer:

The quadrilateral formed by connecting the midpoints of another quadrilateral's sides is a rhombus if the diagonals of the original quadrilateral are perpendicular. This satisfies the properties of a rhombus where the diagonals bisect each other at right angles.

Step-by-step explanation:

The question seeks to determine under what condition the quadrilateral formed by connecting the midpoints of another quadrilateral's sides is a rhombus. One key property of a rhombus is that its diagonals are perpendicular. Thus, if the diagonals of the given quadrilateral PQRS are perpendicular, then the resulting figure from connecting the midpoints in order will indeed be a rhombus. This is because the diagonals of the rhombus formed by the midpoints will be the halves of the diagonals of the original quadrilateral, and due to the Midpoint Theorem, they will also bisect each other at right angles, satisfying the conditions of a rhombus.

The other options given, such as if PQRS is a rhombus, a parallelogram, or if its diagonals are equal, are not necessarily conditions that would result in the midpoints forming a rhombus.

Therefore, the correct answer is (C) if the diagonals of PQRS are perpendicular.

User Shanto George
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