26. A poor artist on the street makes funny cartoons for children and earns his living. Oneehe made a comic face by drawing'a cirele within a cirele, the radius of the bigger circle being30 em and that of smaller being 20 cm as shown in the figure. What is the area of the cap given inthis figure? What qualities of this artist are being reflected here ?Capth ton hy a plane parallel to the

26. A poor artist on the street makes funny cartoons for children and

1.	26. A poor artist on the street makes funny cartoons for children and earns his living. Oneehe made a comic face by drawing'a cirele within a cirele, the radius of the bigger circle being30 em and that of smaller being 20 cm as shown in the figure. What is the area of the cap given inthis figure? What qualities of this artist are being reflected here ?Capth ton hy a plane parallel to the
Answer» LetAandBbethecentresofthesmallerandbiggercircles,respectively.Wehave,BC=BD=BF=BE=30cm,AP=AF=20cmar(∆CDE)=?Let∆CDEbeanisoscelestrianglerepresentingthecapAs,BF=30cm⇒AB+AF=30⇒AB+20=30⇒AB=30−20=10cmSo,BP=AP−AB=20−10=10cm.....(i)Also,BE=30cm⇒BP+PE=30⇒10+PE=30⇒PE=30−10⇒PE=20cm.....(ii)Since,CDisatangenttothesmallercircleSo,BP⊥CD(Tangentisperpendiculartotheradiusatthepointofcontact)Also,CDisachordofthebiggercircleSo,CP=DP(Perpendiculardrawnfromcentretothechordbisectsthechord).....(iii)Now,in∆BPC,usingpythagorastheoremCP^2=BC^2−BP^2⇒CP=302−102−−−−−−−−√[As,BC=30cm,givenandBP=10cm,from(i)]⇒CP=900−100−−−−−−−−√=800−−−√⇒CP=20√2cmSo,CD=2CD[From(iii)]⇒CD=2(202√)=40√2cmNow,ar(∆CED)=12×CD×PE=12×40√2*20⇒ar(∆CED)=400√2cm^2 Hence,theareaofthecapis400√2cm^2.