Question

Parallelogram circumscribing a circle is a _______.

Answer 1

ABCD is a parallelogram such that its sides touches the circle with centre O at P, Q, R and S.

We know that the opposite sides of the parallelogram are equal. Therefore,

AB = CD .....(1)

BC = AD .....(2)

Also, the length of the tangents drawn from an external point to a circle are equal.

∴ AR = AS .....(3)

DR = DQ .....(4)

BP = SB .....(5)

CP = CQ .....(6)

Adding (3), (4), (5) and (6), we get

AR + DR + BP + CP = AS + SB + CQ + DQ

⇒ AD + BC = AB + CD

⇒ 2AD = 2AB [From (1) and (2)]

⇒ AD = AB .....(7)

From (1), (2) and (7), we have

AB = BC = CD = AD

Hence, the parallelogram ABCD is a rhombus.


Parallelogram circumscribing a circle is a _rhombus_.