If a circle touches all the four sides of aparallelogram, the parallel

1.	If a circle touches all the four sides of aparallelogram, the parallelogram is a rhombus.
Answer» Let ABCD be a parallelogram which circumscribes the circle. AP = AS [Tangents drawn from an external point to a circle are equal in length] BP =BQ [Tangents drawn from an external point to a circle are equal in length] CR= CQ [Tangents drawn from an external point to a circle are equal in length] DR = DS [Tangents drawn from an external point to a circle are equal in length] Consider, (AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)AB + CD = AD + BC But AB= CD and BC = AD [Opposite sides of parallelogram ABCD] AB + CD =AD + BC Hence 2AB = 2BC Therefore, AB= BC Similarly, we get AB= DA and DA = CD Thus, ABCD is a rhombus.

If a circle touches all the four sides of aparallelogram, the parallelogram is a rhombus.

Answer»

Let ABCD be a parallelogram which circumscribes the circle.

AP = AS [Tangents drawn from an external point to a circle are equal in length]

BP =BQ [Tangents drawn from an external point to a circle are equal in length]

CR= CQ [Tangents drawn from an external point to a circle are equal in length]

DR = DS [Tangents drawn from an external point to a circle are equal in length]

Consider, (AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ)AB + CD = AD + BC

But AB= CD and BC = AD [Opposite sides of parallelogram ABCD]

AB + CD =AD + BC

Hence 2AB = 2BC

Therefore, AB= BC

Similarly, we get AB= DA and DA = CD

Thus, ABCD is a rhombus.