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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. ThenA. I is true and I implies IIB. II is trueC. III is falseD. I and III are true |
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Answer» Correct Answer - D `"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)` |
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