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Question
A circle touches all sides of a parallelogram. So the parallelogram must be a, ______.
Options
rectangle
rhombus
square
trapezium
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Solution
A circle touches all sides of a parallelogram. So the parallelogram must be a, rhombus.
Explanation:
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 have
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.
