A) \[2:\sqrt{6}:\sqrt{3}+1\]
B) \[\sqrt{2}:\sqrt{6}:\sqrt{3}+1\]
C) \[2:\sqrt{3}:\sqrt{3}+1\]
D) \[3:4:5\]
Correct Answer: A
Solution :
Let the angles of a triangle are 39, 49, 59. We know, \[\angle A+\angle B+\angle C=180{}^\circ \] \[\Rightarrow \] \[3\theta +4\theta +5\theta =180{}^\circ \] \[\Rightarrow \] \[12\theta =180{}^\circ \] \[\Rightarrow \] \[\theta =15{}^\circ \] \[\therefore \]Angles are\[45{}^\circ ,\text{ }60{}^\circ ,\text{ }75{}^\circ \] Now, \[sin\text{ }A=sin\text{ }45{}^\circ =\frac{1}{\sqrt{2}}\] \[sin\,B=sin\,60{}^\circ =\frac{\sqrt{3}}{2}\] \[sin\,C=sin\,75{}^\circ =\frac{\sqrt{3}+1}{2\sqrt{2}}\] \[\therefore \] \[a:b:c=sin\text{ }A:sin\text{ }B:sin\text{ }C\] \[=\frac{1}{\sqrt{2}}:\frac{\sqrt{3}}{2}:\frac{\sqrt{3}+1}{2\sqrt{2}}\]
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