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वृत्त

Question
CBSEHHIMAH10010481

आकृति 7 में, दो चाप PAQ दर्शाई गई हैं। चाप PAQ, O केन्द्र वाले वृत्त का भाग है, जिसकी त्रिज्या OP है तथा चाप PBQ, PQ को व्यास मान कर बनाया गया अर्धवृत्त है जिसका केंद्र M है। यदि OP = 10 सेमी तथा PQ = 10 सेमी तो दर्शाइए कि रेखांकित भाग का क्षेत्रफल 25 left parenthesis square root of 3 space minus space begin inline style straight pi over 6 end style right parenthesis स े म ी squared है। 
                                  

Solution

हम जानते हैं कि,
OP = OQ = 10 cm
PQ = OQ = PQ = 10 cm
therefore space increment POQ space समभ ु ज space ह ै comma
rightwards double arrow space angle POQ space equals space 60 degree
straight D space equals space 70 space cm comma space straight r space equals space 5 space cm comma space straight theta space equals space 60 degree comma space straight R space equals space 10 space cm
छ ा य ां क ि त space क ् ष े त ् र space क ा space क ् ष े त ् रफल space equals space increment OPQ space क ा space क ् ष े त ् रफल space plus space अर ् धव ृ त ् त space क ा space क ् ष े त ् रफल left parenthesis PBQM right parenthesis space minus space क ् ष े त ् र space क ा space क ् ष े त ् रफल left parenthesis OPAQ right parenthesis
equals space begin inline style fraction numerator square root of 3 over denominator 4 end fraction end style left parenthesis straight D right parenthesis squared space plus space begin inline style 1 half end style πr squared space minus space begin inline style fraction numerator straight theta over denominator 360 degree end fraction end style πR squared
equals space begin inline style fraction numerator square root of 3 over denominator 4 end fraction end style left parenthesis 10 right parenthesis squared space plus space begin inline style 1 half end style space straight x space straight pi left parenthesis 5 right parenthesis squared space minus begin inline style fraction numerator 60 degree over denominator 360 degree end fraction end style straight pi left parenthesis 10 right parenthesis squared
equals space begin inline style fraction numerator square root of 3 over denominator 4 end fraction end style left parenthesis 100 right parenthesis space plus space begin inline style 25 over 2 end style straight pi space minus space begin inline style fraction numerator 100 straight pi over denominator 6 end fraction end style
equals space 25 square root of 3 space minus space begin inline style fraction numerator 25 straight pi over denominator 6 end fraction end style
equals space 25 left parenthesis square root of 3 space minus space begin inline style straight pi over 6 end style right parenthesis space cm squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space therefore space space space स ि ध ् द space ह ु आ । space