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 \[3\theta ,\]\[4\theta ,\] \[5\theta \]. We know \[\angle A+\angle B+\angle C={{180}^{o}}\] \[\Rightarrow \] \[3\theta +4\theta +5\theta ={{180}^{o}}\] \[\Rightarrow \] \[12\theta ={{180}^{o}}\] \[\Rightarrow \] \[\theta ={{15}^{o}}\] \[\therefore \] Angles are \[{{45}^{o}},\] \[{{60}^{o}},\]\[{{75}^{o}}\] Now, \[\sin A=\sin {{45}^{o}}=\frac{1}{\sqrt{2}}\] \[\sin \,B=\sin {{60}^{o}}=\frac{\sqrt{3}}{2}\] \[\sin \,C=\sin {{75}^{o}}=\frac{\sqrt{3}+1}{2\sqrt{2}}\] \[\therefore \]\[a:b:c=\sin A:\sin B:\sin C\] \[=\frac{1}{\sqrt{2}}:\frac{\sqrt{3}}{2}:\frac{\sqrt{3}+1}{2\sqrt{2}}\] \[=2:\sqrt{6}:\sqrt{3}+1\]
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