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If the angles of a triangle are in the r...

If the angles of a triangle are in the ratio `4:1:1,` then the ratio of the longest side to the perimeter is

A

`sqrt(3):(2+sqrt(3))`

B

`1 : 6`

C

`1:(2+sqrt(3))`

D

`2 : 3`

Text Solution

Verified by Experts

The correct Answer is:
A
Here , ratio of angles are `4 : 1 : 1`.
`rArr4x+x+x=180^(@)`
`rArrx=30^(@)`
`thereforeangleA=120^(@),angleB=angleC=30^(@)`

Thus ,the ratio of longest side to the perimeter `=(a)/(a+b+c)` Let b = c =x
`thereforea^(2)=b^(2)+c^(2)-2bc cosA`
`rArra^(2)=2x^(2)-2x^(2)cosA=2x^(2)(1-cosA)`
`rArra^(2)=4x^(2)"sin"^(2)(A)/(2)`
`rArra=2x"sin"(A)/(2)`
`rArra=2xsin60^(@)=sqrt(3)x`
Thus ,required ratio is
`(a)/(a+b+c)=(sqrt(3)x)/(x+x+sqrt(3)x)=(sqrt(3))/(2+sqrt(3))`
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