Question

The quadrilateral formed joining the mid-points of the sides of quadrilaterals PQRS, taken in order, is a rhombus, 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 rhombus.
Let the rhombus 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 = $\frac{1}{2}$PR

and In ∆QRS,
BC = $\frac{1}{2}$QS

Since, AB = BC (sides of a rhombus are equal)
Therefore, $\frac{1}{2}$PR = $\frac{1}{2}$QS
⇒ PR = QS
​

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