A) \[\sqrt{3}:2+\sqrt{3}\]
B) \[1:6\]
C) \[1:2+\sqrt{3}\]
D) \[2:3\]
E) \[\sqrt{2}:2+\sqrt{3}\]
Correct Answer: A
Solution :
Let the angles of a triangle are\[4x,\text{ }x\]and\[x\] respectively. \[\therefore \] \[4x+x+x=180{}^\circ \] \[\Rightarrow \] \[x=30{}^\circ \] We know that \[a:b:c\] \[=sin\,A:sin\,B:sin\,C\] \[=sin\text{ }120{}^\circ :\text{ }sin\text{ }30{}^\circ :\text{ }sin\text{ }30{}^\circ \] \[=\frac{\sqrt{3}}{2}:\frac{1}{2}:\frac{1}{2}=\sqrt{3}:1:1\] Let the sides of a triangle are\[\sqrt{3}y,y\]and\[y\]. \[\therefore \]Perimeter of a triangle\[=(\sqrt{3}+1+1)y\] Hence, required ratio \[=\frac{\sqrt{3}y}{(2+\sqrt{3})y}\] \[=(\sqrt{3}):(2+\sqrt{3})\]
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