Question

The quadrilateral formed by joining the mid points of the sides of a quadrilateral PQRS, taken in order, is a rectangle, if diagonals of PQRS are ________.

Answer 1

Given:
PQRS is a quadrilateral
The quadrilateral formed by joining the mid points of the sides of a quadrilateral PQRS, is a rectangle.
Let the rectangle be ABCD.



A is the mid-point of PQ, B is the mid-point of QR, C is the mid-point of RS and D is the mid-point of PS.

Using mid-point theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side and is equal to the half of it.

In ∆PQR,
AB || PR

and In ∆QRS,
BC || QS

Since, AB$\perp$BC
Therefore, PR$\perp$QS


Hence, the quadrilateral formed by joining the mid points of the sides of a quadrilateral PQRS, taken in order, is a rectangle, if diagonals of PQRS are perpendicular.