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If the angles of a triangle are 30^0a n ...

If the angles of a triangle are `30^0a n d45^0` and the included side is `(sqrt(3)+1)c m` then the area of the triangle is______.

Text Solution

Verified by Experts

The correct Answer is:
`(1+sqrt(3))/(2)` sq cm
By sine rule, `(a)/(sinA)=(b)/(sinB)=(c)/(sinC)`
`rArr (sqrt(3)+1)/(sin(105^(@)))=(b)/(30^(@))`

`rArr b=((sqrt(3)+1)sin30^(@))/(sin105^(@))`
`:.` Area of triangle
`(1)/(2)ab sin 45^(@)=(1)/(2)(sqrt(3)+1)((sqrt(3)+1)sin 30^(@)sin 45^(@))/(sin105^(@))`
`=(1)/(2)(sqrt(3)+1)/((sin45^(@)cos60^(@)+cos45^(@)sin60^(@)))*(1)/(2)*(1)/(sqrt(2))`
`=(1)/(4sqrt(2))((3+1+2sqrt(3)))/(((1)/(sqrt(2))*(1)/(2)+(1)/(sqrt(2))*(sqrt(3))/(2)))=((4+2sqrt(3)))/(4sqrt(2)(1+sqrt(3)))*2sqrt(2)`
`=((1+sqrt(3))^(2))/(2(1+sqrt(3)))=(1+sqrt3)/(2)` sq cm
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