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Regular pentagon ABCDE is inscribed in a unit circle. Compute AB*AC.

Guest Feb 4, 2021

CPhill · Answer 1 · 2021-02-04T04:51:27

Using the Law of Cosines

AB = sqrt (1^2 + 1^2 - 2 cos ( 72) ) = sqrt ( (1/2) (5 -sqrt (5) )

AC = sqrt [ AB^2 + AB^2 - 2 (AB^2) cos ( 108) ] = sqrt [ (1/2) (5 + sqrt (5)) ]

AB * AC = sqrt [ (1/2) (5 -sqrt (5)) (1/2) (5 + sqrt (5)) ] = (1/2) sqrt [ 25 - 5 ] = (1/2) sqrt (20) =

(1/2) 2 sqrt (5) = sqrt (5)