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Question Number 195611 by Mingma last updated on 05/Aug/23

Answered by mr W last updated on 06/Aug/23

a=side length of small pentagon b=side length of big pentagon b=2a sin 54° a=2r sin 36° ⇒r=(a/(2 sin 36°)) b=2R sin 36° ⇒R=(b/(2 sin 36°))=((a sin 54°)/(sin 36°)) total area =((5 R^2 sin 72°)/2)+((a^2 sin 72°)/2) shaded area=((5 r^2 sin 72°)/2)−((a^2 sin 72°)/2) ((shaded)/(total))=((5r^2 −a^2 )/(5R^2 +a^2 ))=((5((r/a))^2 −1)/(5((R/a))^2 +1)) =((5((1/(2 sin 36°)))^2 −1)/(5(((sin 54°)/(sin 36°)))^2 +1))=(1/4)=25% ⇒answer b)

a=sidelengthofsmallpentagonb=sidelengthofbigpentagonb=2asin54°a=2rsin36°⇒r=a2sin36°b=2Rsin36°⇒R=b2sin36°=asin54°sin36°totalarea=5R2sin72°2+a2sin72°2shadedarea=5r2sin72°2−a2sin72°2shadedtotal=5r2−a25R2+a2=5(ra)2−15(Ra)2+1=5(12sin36°)2−15(sin54°sin36°)2+1=14=25%⇒answerb)

Commented by Mingma last updated on 06/Aug/23

Perfect ��

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