A) \[2:3:5\]
B) \[1:2:3\]
C) \[1:3:7\]
D) \[3:7:9\]
Correct Answer: B
Solution :
We have, \[\Delta =\frac{\sqrt{3}}{4}{{a}^{2}},s=\frac{3a}{2}\] \[\therefore \]In radius \[r=\frac{\Delta }{s}=\frac{a}{2\sqrt{3}}\] Circumradius \[R=\frac{abc}{4\Delta }=\frac{{{a}^{3}}}{\sqrt{3}{{a}^{2}}}=\frac{a}{\sqrt{3}}\] and exradii \[{{r}_{1}}=\frac{\Delta }{s-a}=\frac{\sqrt{3}\text{/}4{{a}^{2}}}{a\text{/}2}\] \[=\frac{\sqrt{3}}{2}a\] \[\therefore \]Required ratio\[=r:R:{{r}_{1}}\] \[=\frac{a}{2\sqrt{3}}:\frac{a}{\sqrt{3}}:\frac{\sqrt{3}}{2}a\] \[=1:2:3\]
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