Question

Four alternative answers for the following question is given. Choose the correct alternative.

A circle touches all sides of a parallelogram. So the parallelogram must be a, ______

Options

• rectangle 

• rhombus

• square 

• trapezium 

Answer 1

ABCD is a parallelogram. A circle with centre O touches the parallelogram at E, F, G and H.

ABCD is a parallelogram.
∴ AB = CD     .....(1)       (Opposite sides of parallelogram are equal)
AD = BC         .....(2)       (Opposite sides of parallelogram are equal)
Tangent segments drawn from an external point to a circle are congruent.
AE = AH           .....(3)
DG = DH          .....(4)
BE = BF            .....(5)
CG = CF           .....(6)
Adding (3), (4), (5) and (6), we get
AE + BE + CG + DG = AH + DH + BF + CF
⇒ AB + CD = AD + BC      .....(7)
From (1), (2) and (7), we ahve
2AB = 2BC
⇒ AB = BC           .....(8)
From (1), (2) and (8), we have
AB = BC = CD = AD
∴ Parallelogram ABCD is a rhombus.                   (A rhombus is a parallelogram with all sides equal)
Hence, the correct answer is rhombus .