Let A1 ,A2, An be the vertices of an n-sided regular polygon such that (1/ A1 A2 ) = (1/ A1 A3 ) + (1/ A1 A4 ) . Find the value of n.

Question

Let A1 ,A2, An be the vertices of an n-sided regular polygon such that (1/ A1 A2 ) = (1/ A1 A3 ) + (1/ A1 A4 ) . Find the value of n. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Let O be the centre and r be the radius of the circle
passing through the vertices t

A_{1}, A_{2}, A_{n}

,
Then,

∠ A_{1} O A_{2} = \frac{2 π}{n}, a l so O A_{1} = O A_{2} = r

Again, by cos formula, we know that,
cos

(\frac{2 π}{n}) = \frac{O A _{1}^{2} + O A _{2}^{2} - A _{1} A _{2}^{2}}{2 ( O A _{1} ) ( O A _{2} )}

\Rightarrow cos (\frac{2 π}{n}) = \frac{r ^{2} + r ^{2} - A _{1} A _{2}^{2}}{2 ( r ) ( r )}

\Rightarrow 2 r^{2} cos (\frac{2 π}{n}) = 2 r^{2} - A_{1} A_{2}^{2}

\Rightarrow A_{1} A_{2}^{2} = 2 r^{2} - 2 r^{2} cos (\frac{2 π}{n})

\Rightarrow A_{1} A_{2}^{2} = 2 r^{2} [1 - cos (\frac{2 π}{n})]

\Rightarrow A_{1} A_{2}^{2} = 2 r^{2} .2 s i n^{2} (\frac{π}{n})

\Rightarrow A_{1} A_{2}^{2} = 4 r^{2} s i n^{2} (\frac{π}{n})

\Rightarrow A_{1} A_{2} = 2 r s in (\frac{π}{n})

Similarly,

A_{1} A_{3} = 2 r s in (\frac{2 π}{n})

and

A_{1} A_{4} = 3 r s in (\frac{3 π}{n})

Since,

\frac{1}{A _{1} A _{2}} = \frac{1}{A _{1} A _{3}} + \frac{1}{A _{1} A _{4}} .

[given]

\Rightarrow \frac{1}{2 r s in ( π / n )} = \frac{1}{2 r s in ( 2 π / n )} + \frac{1}{2 r s in ( 3 π / n )}

\Rightarrow \frac{1}{s in ( π / n} = \frac{1}{s in ( 2 π / n} + \frac{1}{s in ( 3 π / n}

\Rightarrow \frac{1}{s in ( π / n} = \frac{s in ( \frac{3 π}{n} ) + s in ( \frac{2 π}{n} )}{s in ( 2 π / n ) ( s in ( 3 π / n )}

\Rightarrow s in (\frac{2 π}{n}) . s in (\frac{3 π}{n}) = s in (\frac{π}{n}) s in (\frac{3 π}{n}) + s in (\frac{π}{n}) . s in (\frac{2 π}{n})

\Rightarrow s in (\frac{2 π}{n}) [s in (\frac{3 π}{n}) - s in (\frac{π}{n})] = s in (\frac{π}{n}) . s in (\frac{3 π}{n})

\Rightarrow s in (\frac{2 π}{n}) [{2 cos (\frac{3 π + π}{2 n}) s in (\frac{3 π - π}{2 n})}] = s in (\frac{π}{n}) . s in (\frac{3 π}{n})

\Rightarrow 2 s in (\frac{2 π}{n}) . cos (\frac{2 π}{n}) . s in (\frac{π}{n}) = s in (\frac{π}{n}) s in (\frac{3 π}{n})

\Rightarrow 2 s in (\frac{2 π}{n}) cos (\frac{2 π}{n}) = s in (\frac{3 π}{n})

\Rightarrow s in (\frac{4 π}{n}) = s in (\frac{3 π}{n})

\Rightarrow \frac{4 π}{n} = π - \frac{3 π}{n}

\Rightarrow \frac{7 π}{n} = π

\Rightarrow n = 7

.

Q. Let A1​,A2​,An​ be the vertices of an n-sided regular polygon such that A1​A2​1​=A1​A3​1​+A1​A4​1​. Find the value of n.

Q. Let $A_{1}, A_{2}, A_{n}$ be the vertices of an n-sided regular
polygon such that $\frac{1}{A _{1} A _{2}} = \frac{1}{A _{1} A _{3}} + \frac{1}{A _{1} A _{4}} .$ Find the value of n.