Question

A poor artist on the street makes funny cartoons for children and earns his living. Once he made a comic face by drawing a circle within a circle, the radius of the bigger circle being 30 cm and that of smaller being 20 cm as shown in figure. What is the area of the cap give in this figure? What qualities of this artist are being reflected here?

Answer 1

Let A and B be the centres of the smaller and bigger circles,respectively.We have,BC=BD=BF=BE=30 cm,AP=AF=20 cmar(∆CDE)=?Let ∆CDE be an isosceles triangle representing the capAs, BF=30 cm⇒AB+AF=30⇒AB+20=30⇒AB=30−20=10 cmSo, BP=AP−AB=20−10=10 cm        .....(i)Also, BE=30 cm⇒BP+PE=30⇒10+PE=30⇒PE=30−10⇒PE=20 cm       .....(ii)Since, CD is a tangent to the smaller circleSo, BP⊥CD          (Tangent is perpendicular to the radius at the point of contact)Also, CD is a chord of the bigger circleSo, CP=DP          (Perpendicular drawn from centre to the chord bisects the chord)      .....(iii)Now, in ∆BPC, using pythagoras theoremCP2=BC2−BP2⇒CP=302−102−−−−−−−−√          [As, BC=30 cm, given and BP=10 cm, from (i)]⇒CP=900−100−−−−−−−−√=800−−−√⇒CP=202√ cmSo, CD=2CD               [From (iii)]⇒CD=2(202√)=402√ cmNow, ar(∆CED)=12×CD×PE=12×402√×20⇒ar(∆CED)=4002√ cm2Hence, the area of the cap is 4002√ cm2. 

Hope  this information  will clear your doubts  about the topic.
If you have any more doubts just ask here on the forum and our experts  will  try to help you out as soon as possible.Regards

Answer 2

Answer:

Step-by-step explanation:

Radius of bigger triangle is with centre O=30 cm

Radius from O'=20cm

•

••, difference of their radii =distance between their centres O andO' =10cm

AB is a tangent if smaller circle so,/_OCA=90°=/_OCB

In triangle ica

AO²=AC²+CO²

30²=AC²+10²

900=AC²+100

AC=20√2

AC=CB

AB=AC+CB

AB=AC+CB

AB=2AC

AB=2*20√2

AB=40√2cm

CD=30-10

=20cm

Area of cap=AR of adb

1/2×AB×CD

1/2×40√2×20=400√2cm²

400×1.414

=565.6cm²