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If the angles of triangle are in the ...

If the angles of 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

`1:3:2`

C

`1:2+sqrt(3)`

D

`2:3`

Text Solution

Verified by Experts

The correct Answer is:
A
Given, ratio of angles are `4:1:1`,
`rArr 4x+x+x=180^(@)`
`rArr x=30^(@)`
`:. angleA=120^(@), angleB=angleC=30^(@)`

Thus, ratio of longest side to perimeter `=(a)/(a+b+c)`
Let `b=c=x`
`rArr a^(2)=b^(2)+c^(2)-2abc cos A` [ by consie rule]
`rArr a^(2)=2x^(2)-2x^(2)cosA`
`=2x^(2)(1-cosA)`
`rArr a^(2)=4x^(2)sin^(2)A//2`
`rArra^(2)=2xsin A//2`
`rArra=2x sin 60^(@)=sqrt(3)x`
Thus, required ratio
`=(a)/(a+b+c)`
`=(sqrt(3x))/(x+x+sqrt(3)x)`
`=(sqrt(3))/(2+sqrt(3))`
`sqrt(3):2+sqrt(3)`
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