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Question

A restaurant POQR is being constructed in the middle of a circular lake. The circle, which has its center at O is touching the mid-points of the boundary ABCD as shown in the figure. If AB=CD and AD=BC, then

Quadrilateral ABCD can be

A
rhombus
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B
square
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C
rectangle
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D
parallelogram
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Solution

The correct option is B square
Given, a circle is inscribed in a quadrilateral, such that AB,BC,CD and DA acts as tangents to one circle.
AB+CD=BC+AD2AB=2BCAB=BC=CD=AD
In quadrilateral OQAP,
OQA=OPA=90PAQ=90
[radius is perpendicular to the tangent at the point of contact]
ADC=DCB=CBA=BAD=90
Hence, ABCD is a square.

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