question_answer

Let \[\Delta ABC\] be an equilateral triangle with side length a. For any point P inside \[\Delta ABC\], let S' denote the sum of the distances of P from each of the sides of \[\Delta ABC\]. The difference between the maximum and minimum values of S is

A) \[a/3\]                  

B)       \[a/2\]                   

C) \[2a/3\]                 

D)       None of these

Correct Answer: D

Solution :

[d] \[ar(\Delta ABC)=ar(\Delta BPC)+ar(\Delta CPA)+ar(\Delta APB)\]\[\Rightarrow \,\,\,\Delta =\frac{1}{2}ax+\frac{1}{2}ay+\frac{1}{2}az\] \[\Rightarrow \,\,\,\Delta =\frac{a}{2}(a+y+z)\] \[\Rightarrow \,\,\,\,\,\,S=x+y+z=\frac{2\Delta }{a}=\text{constant}\] Therefore, required difference is zero.