A) \[\frac{3}{2+\sqrt{3}}\]
B) \[\frac{1}{2+\sqrt{3}}\]
C) \[\frac{\sqrt{3}}{\sqrt{3}+2}\]
D) \[\frac{2}{2+\sqrt{3}}\]
Correct Answer: C
Solution :
| [c] Consider a triangle ABC. |
| Given, angles of a triangle are in the ratio \[4:1:1.\] angles are 4x, x and x |
| i.e., \[\angle A=4x,\angle B=x,\angle C=x\] |
| Now, by angle sum property of\[\Delta \], we have |
| \[\angle A+\angle B+\angle C=180{}^\circ \] |
| \[\Rightarrow 4x+x+x=180{}^\circ \Rightarrow x=\frac{180{}^\circ }{6}=30{}^\circ \] |
| \[\therefore \angle A=120{}^\circ ,\angle B=30{}^\circ ,\angle C=30{}^\circ \] |
| We know, ratio of sides of \[\Delta ABC\]is given by \[\sin A:\sin B:\operatorname{sinC}=sin120{}^\circ :sin30{}^\circ :sin30{}^\circ \] |
| \[=\frac{\sqrt{3}}{2}:\frac{1}{2}:\frac{1}{2}=\sqrt{3}:1:1\] |
| Required ratio \[=\frac{\sqrt{3}}{1+1+\sqrt{3}}=\frac{\sqrt{3}}{2+\sqrt{3}}.\] |
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