Let `A_(1),A_(2),A_(3),.........,A_(14)` be a regular polygon with 14 sides inscribed in a circle of radius R. If `(A_(1)A_(3))^(2)+(A_(1)A_(7))^(2)+(A_(3)A_(7))^(2)=KR^(2)`, then K is equal to :

Answer

7

Explanation

`sin.(pi)/(7)=((1)/(2)A_(1)A_(3))/(R)impliesA_(1)A_(3)=2Rsin.(pi)/(7)`
similarly `A_(1)A_(7)=2Rsin.(3pi)/(7)" and " A_(3)A_(7)=2R sin. (2pi)/(7)`
`therefore (A_(1)A_(3))^(2)+(A_(1)A_(7))^(2)+(A_(3)A_(7))^(2)=4R^(2)(sin^(2).(pi)/(7)+sin^(2).(2pi)/(7)+sin^(2).(3pi)/(7))`
`=4R^(2)xx(7)/(4)=7R^(2)`