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Question

Inradius of a circle which is inscribed in an isosceles triangle one of whose angle is 2π3, is 3, then the area of the triangle is


A

43

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B

1273

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C

12+73

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D

None of these

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Solution

The correct option is C

12+73


Explanation For Correct Option:

Finding the Area Of The Triangle:

The given triangle is an isosceles triangle.

Let A=2π3=120°,b=c

Then B=C=30°[bypropertyofisoscelestriangle]
From sine rule

sin120°a=sin30°b

32a=12bsin30°=12&sin120°=32a=3b...(i)

Area of triangle

=bcsinA2=3b24.....(ii)[from(i)]

Inradius of a circle

r=ss=a+b+c2

s=3[givenr=3]3b24s=3[from(ii)]b2=4s=2a+4bs=a+b+c2,b=cb2=23b+4bb=23+4

Substituting the value of b in equation (ii)

=3b24=3(23+4)24=12+73

Hence, option (C) is the correct answer.


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