Question

The points of contact of the tangents from an exterior point P to the circle with centre O
are A and B. If m∠OPB = 30, then m∠AOB = ......,select a proper option (a), (b), (c) or (d) from given options so that the statement becomes correct.
(a) 30
(b) 60
(c) 90
(d) 120

Answer 1

We know, tangent of circle at point of contact making right angle with the radius.

In any circle if we draw two tangent from the same exterior point then tangent are symetrical in length.

Hence, ∠OPB = ∠OPA.

Hence, ∠APB = ∠OPB+∠OPA=60°

Now, In quadrilateral AOPB

∠OAP+∠APB+∠PBO+∠AOB = 360°

We know, ∠OAP=∠PBO=90°

Hence,90°+60°+90°+∠AOB = 360°

So, ∠AOB = 120°.

Option d is correct.

Answer 2

Option ( d ) is correct .

Explanation :

We know that ,

<APO = <OPB = 30°

Therefore ,

<APB =2 × <APO = 2×30° = 60°

Now ,

In OAPB Quadrilateral ,

<OAP = = OBP = 90°

<AOB + <APB = 180°

=> <AOB + 60° = 180°

=> <AOB = 180° - 60°

=> <AOB = 120°

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