Let A 1 , A 2 … .. A n be vertices of an n sided regular polygon such that A 1 A 2 1 = A 1 A 3 1 + A 1 A 4 1 . Then find values of n

Let A 1 ( Z 1 ) , A 2 ( Z 2 ) , A 3 ( Z 3 ) , A 4 ( Z 4 ) A 1 A 2 = ∣ Z 1 ∣ 2 sin n π , A 1 A 3 = ∣ Z 1 ∣ 2 sin n 2 π , A 1 A 4 = ∣ Z 1 ∣ 2 sin n 3 π ⇒ s i n n π 1 = s i n n 2 π 1 + s i n n 3 π 1 ⇒ n = 7

Mathematics - Complex Number Question with Solution | TestHub

MathematicsComplex NumberGeometric ApplicationMedium2 minQB

MathematicsMediuminteger

Let $A_{1}, A_{2} \dots .. A_{n}$ be vertices of an n sided regular polygon such that $\frac{1}{A _{1} A _{2}} = \frac{1}{A _{1} A _{3}} + \frac{1}{A _{1} A _{4}}$ . Then find values of n

Answer:

Solution:

Let $A_{1} (Z_{1}), A_{2} (Z_{2}), A_{3} (Z_{3}), A_{4} (Z_{4})$ $A_{1} A_{2} = ∣ Z_{1} ∣ 2 sin \frac{π}{n}, A_{1} A_{3} = ∣ Z_{1} ∣ 2 sin \frac{2 π}{n}, A_{1} A_{4} = ∣ Z_{1} ∣ 2 sin \frac{3 π}{n}$ $\Rightarrow \frac{1}{s i n \frac{π}{n}} = \frac{1}{s i n \frac{2 π}{n}} + \frac{1}{s i n \frac{3 π}{n}} \Rightarrow n = 7$

Stream:JEESubject:MathematicsTopic:Complex NumberSubtopic:Geometric Application

⏱ 2mℹ️ Source: QB

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