Question

Question 9
The figure obtained by joining the mid-points of the sides of a rhombus, taken In order, is
(A) a rhombus
(B) a rectangle
(C) a square
(D) any parallelogram

Answer 1

Let ABCD be a rhombus in which P,Q,R and S are the mid-points of sides AB,BC,CD and DA, respectively.

Join AC , PR and SQ

In $\Delta ABC$ P is the mid-point of AB and Q is the mid-point of BC.

$\Rightarrow PQ\parallel ACandPQ=\frac{1}{2}AC$ [ by using mid-point theorem]……(i)

Similarly, in DAC,

SR ∥ AC and $SR=\frac{1}{2}AC$ …..(ii)

From Eq (i) and (ii)

PQ ∥ SR and PQ = SR

So, PQRS is a parallelogram,

Also ABQS is a parallelogram.

$\Rightarrow AB=SQ$ …(iii)

[ Opposite sides of a parallelogram are equal]

Similarly, PBCR is a parallelogram

$\Rightarrow BC=PR$ [ Opposite sides of Parallelogram are equal]
$\Rightarrow AB=PR[\because BC=AB$ sides of a rhombus]

$\Rightarrow SQ=PR$

So, the diagonal of a parallelogram are equal.

Hence, PQRS is a rectangle