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The picture shows a regular pentagon ABCDE. Find the ratio of the area of triangle ABD to the area of pentagon ABCDE.

 

 Jan 14, 2021
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Let the side of the pentagon  =   S

 

Area of  pentagon    ≈  1.72 S^2

 

Height of triangle  

 

tan 72°   =  H  / (1/2 S)

 

H  =       S tan 72° / ( 2)

 

Area of triangle  =   (1/2) S  *  ( S  tan 72°) / ( 2)   =  S^2 tan 72°/4   

 

Ratio    area of triangle to  area  of pentagon  =       (tan (72°) / 4 )  /   1.72  =    

 

tan (72°) / ( 4 * 1.72)   ≈   .447

 

 

cool cool cool

 Jan 14, 2021

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