Let A₁ A₂ A₃ A₉ be a nine-sided regular polygon with side length 2 units. The difference between the lengths of the diagonals A ₁ ~A ₅ and A ₂ ~A ₄ equals

Step-by-step solution

A ₁ OA ₂= 2 9 =40^ 2 9 = x²+x²-4 2 x² .x² 40^ )=x²-2x² (1- 40^ )=2x= 1 20^ .........(i) Now 8 9 = x²+x²- (A₁ A₅ )² 2 x² (in . A₁ O A₅ ) array l (A₁ A₅ )²=2 x² (1- 8 9 ) =2 x² (1- 160^ ) =4 x² ² 80^ array A ₁ ~A ₅=2 x 80^ .........(ii) Similarly in A ₂ OA ₄ ~A ₂ ~A ₄=2 x 40^ .........(iii) (ii) - (iii) aligned A ₁ ~A ₅- A ₂ ~A ₄ &=2 x ( 80^ - 40^ ) &=2( using ( i )) aligned