In the given figure, AR and RT are tangents to the circle with centre O. B is the mid-point of QS. If AOBT is a straight line, then which of the following is/are true? T B S А P R. (1) OA = OR (2) OQTS is a cyclic quadrilateral (3) SQ = AR (4) All of these.
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AR and RT are tangents to the circle with center O.
QB=SB, AOBT is a straight line.
Now, join QT and OS.
QT is a tangent and OS is the radius of the circle.
Since, tangents are perpendicular to radii at the point of contact.
∴ OS⊥TR and OQ⊥QT
⇒ ∠OST=∠OQT=90⁰.
In quadrilateral OQTS
∠OST=∠OQT=90⁰
⇒ ∠OST+∠OQT=180⁰-- ( 1 )
Also, ∠OST+∠STQ+∠TQO+∠QOS=360⁰ [ Sum of properties of a quadrilateral ]
⇒ ∠STQ+∠QOS=180⁰ [ From ( 1 ) ]
In a cyclic quadrilateral, the sum of the opposite angles is 180⁰
This property is satisfied by quadrilateral OQTS
∴ OQTS is a cyclic quadrilateral
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