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
• B. square
• C. rectangle
• D. parallelogram

Answer 1

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.
$\Rightarrow AB+CD=BC+AD\Rightarrow 2AB=2BC\Rightarrow AB=BC=CD=AD$
In quadrilateral OQAP,
$\angle OQA=\angle OPA={90}^{\circ }\Rightarrow \angle PAQ={90}^{\circ }$
[radius is perpendicular to the tangent at the point of contact]
$\therefore \angle ADC=\angle DCB=\angle CBA=\angle BAD={90}^{\circ }$
Hence, ABCD is a square.