Question

ABCD is a rectangle and P, Q, R and S are mid-points of the side AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

Answer 1

Join DB and AC such that two triangles ADB and DBC are formed.

We know that the line joining the midpoints of the two sides of triangle is parallel to the third side and is of half the measure of third side.

Here in triangle ADB P and S are the midpoints of AB and AD respectively,hence $PS\parallel DB$ also $PS=\frac{1}{2}DB$..................1

Similarly,$RQ\parallel DB$ also $RQ=\frac{1}{2}DB$..................2

$PQ\parallel AC$ also $PQ=\frac{1}{2}AC$....................3

$SR\parallel AC$ also $SR=\frac{1}{2}AC$......................4

From equation 1 and 2 we get $PS\parallel RQ$ and $PS=RQ$

And from equation 3 and 4 we get $PQ\parallel SR$ and $PQ=SR$

Therefore, PQRS is a rhombus.