Let `nge4` be a positive integer and let `l_(1),l_(2),.....,l_(n)` be the lengths of the sides of arbitrary n-sided non-degenerate polygon P. Suppose `l_(1)/(l_(2))+l_(2)/(l_(3))+....l_(n-1)/(l_(n))+l_(n)/(l_(1))=n.` Consider the following statements: I. The lengths of the sides of P are equal. II. The angles of P are equal. III. P is a regular polygon if it is cyclic. Then

Options

A. I is true and I implies II
B. II is true
C. III is false
D. I and III are true

Answer

D

Explanation

`"given ":l_(1)/(l_(2))+l_(2)/(l_(3))......+l_(n)/(l_(1))=n.....(i)`
`therefore"Use A.M"geG.M`
We get
`((l_(1)/(l_(2))+l_(2)/(l_(3)).....+l_(n)/(l_(1))))/(n)geroot(n)(l_(1)/(l_(2)).l_(2)/(l_(3))....l_(n)/(l_(1)))`
`therefore(n)/(n)ge1`
`rArrn=n`
So A.M=G.M
`"Hence "l_(1)/(l_(2))=l_(2)/(l_(3))......=l_(n)/(l_(1))=k`
`rArrk=(l_(1)+l_(2).....+l_(n))/(l_(2)+l_(3).....+l_(n)+l_(1))=1`
`l_(1)=l_(2).....=l_(n)`