question_answer

In \[\Delta ABC,\] circumradius is 3 and inradius is 1. Let I be the in centre of the triangle. If D, E and F are the feet of the perpendiculars from I to BC, CA and AB, respectively, then the value of \[\frac{IA.IB.IC}{ID.IE.IF}=\]

A) \[6\]                      

B)        \[9\]                      

C) \[12\]       

D)        \[18\]

Correct Answer: C

Solution :

[c] \[\frac{IA.IB.IC}{ID.IE.IF}=\frac{\left( r\cos ec\frac{A}{2} \right)\left( r\cos ec\frac{B}{2} \right)\left( r\cos ec\frac{C}{2} \right)}{{{r}^{3}}}\] \[=\frac{1}{\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}}=\frac{4R}{r}=12\]