Question

PA, PB touch ⦿ (O,r) at A and B. If m∠AOB = 80, then m∠OPB = .....,select a proper option (a), (b), (c) or (d) from given options so that the statement becomes correct.
(a) 80
(b) 50
(c) 10
(d) 100

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+∠OPB=2∠OPB

Now, In quadrilateral AOPB

∠OAP+∠APB+∠PBO+∠AOB = 360°

We know, ∠OAP=∠PBO=90°

Hence,90°+2∠OPB+90°+80° = 360°

So, 2∠OPB = 100°

∠OPB=50°

option b is correct.

Answer 2

Option ( b ) is correct .

Explanation :

In AOBP Quadrilateral ,

<AOP = <OBP = 90°

[ tangent , radius relation ]

<AOB = 80° [ Given ]

<AOB + <APB = 180°

=> 80° + <APB = 180°

=> <APB= 180° - 80°

=> <APB = 100°

Now ,

<OPB = ( <APB )/2

= 100°/2

= 50°

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