Find the polygon

Geometry Level 4

1 A 1 A 2 = 1 A 1 A 3 + 1 A 1 A 4 \large \frac1{A_1 A_2} = \frac1{A_1 A_3} + \frac1{A_1 A_4 }

If A 1 , A 2 , A 3 , … , A n A_1, A_2, A_3, \ldots, A_n are consecutive vertices of a regular n n -sided polygon such that the equation above is fulfilled, find the value of n n .


The answer is 7.

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Tanishq Varshney by Tanishq Varshney , 23, India

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1 solution

Tanishq Varshney
Jul 16, 2015

O A 1 = O A 2 = O A 3 = r OA_{1}=OA_{2}=OA_{3}=r

< A 1 O A 2 = < A 2 O A 3 = < A 3 O A 4 = 2 π n <A_{1}OA_{2}=<A_{2}OA_{3}=<A_{3}OA_{4}=\frac{2\pi}{n}

( A 1 A 2 ) 2 = r 2 + r 2 − 2 r 2 cos ⁡ ( α ) (A_{1}A_{2})^2=r^2+r^{2}-2r^{2}\cos (\alpha)

where α = 2 π n \alpha=\frac{2\pi}{n}

A 1 A 2 = 2 r sin ⁡ ( α 2 ) \large{A_{1}A_{2}=2r\sin(\frac{\alpha}{2})}

A 1 A 3 = 2 r sin ⁡ ( 2 α 2 ) \large{A_{1}A_{3}=2r\sin(\frac{2\alpha}{2})}

A 1 A 4 = 2 r sin ⁡ ( 3 α 2 ) \large{A_{1}A_{4}=2r\sin(\frac{3\alpha}{2})}

Now substitute the corresponding values

1 sin ⁡ ( α 2 ) = 1 sin ⁡ ( α ) + 1 sin ⁡ ( 3 α 2 ) \large{\color{#3D99F6}{\frac{1}{\sin(\frac{\alpha}{2})}=\frac{1}{\sin(\alpha)}+\frac{1}{\sin(\frac{3\alpha}{2})}}}

Nowadays this property is in fashion \color{#D61F06} {\text{Nowadays this property is in fashion}}

sin ⁡ 3 x sin ⁡ x = 1 + 2 cos ⁡ 2 x \large{\frac{\sin 3x}{\sin x}=1+2\cos 2x}

( 1 + 2 cos ⁡ α ) ( sin ⁡ α ) = sin ⁡ ( 3 α 2 ) + sin ⁡ ( α ) \large{(1+2\cos \alpha)(\sin \alpha)=\sin(\frac{3 \alpha}{2})+\sin(\alpha)}

sin ⁡ ( 2 α ) − sin ⁡ ( 3 α 2 ) = 0 \large{\sin (2\alpha)-\sin(\frac{3 \alpha}{2})=0}

cos ⁡ ( 7 α 4 ) sin ⁡ ( α 4 ) = 0 \large{\cos(\frac{7\alpha}{4})\sin(\frac{ \alpha}{4})=0}

It is clear that π 2 n ≠ 0 \frac{\pi}{2n}\neq 0

hence 7 α 4 = π 2 \large{\frac{7\alpha}{4}=\frac{\pi}{2}}

n = 7 \large{\boxed{n=7}}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

NIce question and solution, Tanishq. For sake of clarity it might be a good idea to specify that the vertices as listed are indeed consecutive, and that we are dealing with regular n-sided polygons.

Brian Charlesworth - 5 years, 11 months ago

Nice solution

Sayandeep Ghosh - 5 years, 1 month ago

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