Question

In the figure, PA and PB are tangents to a circle with centre 0. If angle AOB = 120°, then find
angle OPA.
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Answer 1

Answer:

hey here is your answer

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pls refer the reference diagram uploaded above for better understanding

so here we go

Step-by-step explanation:

now here tangent pa=tangent pb

because tangent segments drawn from external to circle are equal in length

so now in quadrilateral pabo

as one pair of adjacent sides are equal,

quadrilateral pabo is a kite

so thus as it is a kite diagonals ao and ab will bisect the opposite angles

therefore angle opa=opb=half of angle apb (1)

now here angle oap=90

and angle obp=90 (tangent theorem ) (2)

now also angle aob=120

consider quadrilateral pabo

using angle sum property we get

angle obp+oap+aob+apb=360

ie 90+90+120+angle apb=360 (from 2)

ie angle apb=360-300

so angle apb=60

therefore angle opa=1/2×60 (from (1) )

so angle opa= 30 degree

hence value of angle opa is 30 degrees