Let n- 4 be a positive integer and let l1- l2- ln be the lengths of the sides of arbitrary n-sided non-degenerate polygon P- Suppose l1-l2-l2-l3-ln-1-ln-ln-l1-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 cycle- Then-

The correct option is B I and III are trueGiven equation l_1/l_2 + #160; l_2/l_3 #160;+ #160; l_3/l_4 .......+ l_n/l_1 = n From A ...

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Byju's Answer

Standard XII

Physics

Constrained Motion : General Approach

Let n≥ 4 be...

Question

Let $n \geq 4$ 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 $\frac{l_{1}}{l_{2}} + \frac{l_{2}}{l_{3}} + . . . . + \frac{l_{n - 1}}{l_{n}} + \frac{l_{n}}{l_{1}} = n$ . 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 cycle. Then.

I is true and I implies II

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II is true

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III is false

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I and III are true

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Solution

The correct option is B I and III are true
Given equation $\frac{l_{1}}{l_{2}}$ + $\frac{l_{2}}{l_{3}}$ + $\frac{l_{3}}{l_{4}}$ .......+ $\frac{l_{n}}{l_{1}} = n$
From $A . M - G . M$ inequality, we get $\frac{l_{1}}{l_{2}}$ + $\frac{l_{2}}{l_{3}}$ + $\frac{l_{3}}{l_{4}}$ .......+ $\frac{l_{n}}{l_{1}}$ $\geq$ $n$ inequality holds only if all the elements are equal i.e, $\frac{l_{1}}{l_{2}}$ = $\frac{l_{2}}{l_{3}}$ = $\frac{l_{3}}{l_{4}}$ ....... = $\frac{l_{n}}{l_{1}}$ , which gives that $l_{1} = l_{2} = l_{3} . . . . = l_{n}$ so lengths of all sides are equal.
We can not say anything about angles of P.
Now if $P$ is cyclic and as all sides are of equal length, then from triangles formed by center of circle to vertices will have same vertex angle. Therefore $P$ is regular if it is cycle.

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