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+AD⇒2AB=2BC⇒AB=BC=CD=AD In quadrilateral OQAP, ∠OQA=∠OPA=90∘⇒∠PAQ=90∘ [radius is perpendicular to the tangent at the point of contact] ∴∠ADC=∠DCB=∠CBA=∠BAD=90∘ Hence, ABCD is a square.