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वृतों से संबंधित क्षेत्रफल

Question
CBSEHHIMAH10010480

आकृति 6 में, बनी incrementABC, जिसके शीर्ष A के निर्देशांक (0, -1) हैं तथा भुजाओं AB तथा AC के मध्य-बिंदुओं D तथा E के निर्देशांक क्रमशः (1, 0) तथा (0, 1) हैं। यदि F भुजा BC का मध्य-बिंदु हैं तो त्रिभुजों DEF तथा ABC के क्षेत्रफल ज्ञात कीजिए। 
                  

Solution
म ा न ा space straight B space left parenthesis straight p comma space straight q right parenthesis space और space straight C space left parenthesis straight r comma space straight s right parenthesis
AB space क े space मध ् य space ब िं द ु space equals space straight D space क ा space समन ् वय
left parenthesis begin inline style fraction numerator 0 space plus space straight p over denominator 2 end fraction end style comma space begin inline style fraction numerator negative 1 space plus space straight q over denominator 2 end fraction end style right parenthesis space equals space left parenthesis 1 comma space 0 right parenthesis
rightwards double arrow space begin inline style straight p over 2 end style space equals space 1
rightwards double arrow space straight p space equals space 2
rightwards double arrow space begin inline style fraction numerator negative 1 space plus space straight q over denominator 2 end fraction end style space equals space 0
rightwards double arrow space minus 1 space plus space straight q space equals space 0
rightwards double arrow space straight q space equals space 1
therefore space straight B left parenthesis 2 comma space 1 right parenthesis
AC space क े space मध ् य space ब िं द ु space equals space straight E space क ा space समन ् वय
left parenthesis begin inline style fraction numerator 0 space plus space straight r over denominator 2 end fraction end style comma space begin inline style fraction numerator negative 1 space plus space straight s over denominator 2 end fraction end style right parenthesis space equals space left parenthesis 0 comma space 1 right parenthesis
rightwards double arrow space begin inline style straight r over 2 end style space equals 0
rightwards double arrow space straight r space equals space 0
rightwards double arrow space begin inline style fraction numerator negative 1 space plus space straight s space over denominator 2 end fraction end style space equals space 1
rightwards double arrow space minus 1 space plus straight s space equals 2 space
rightwards double arrow space straight s space equals 3
therefore space straight C left parenthesis 0 comma space 3 right parenthesis
straight F space क ा space समन ् वय space equals space BC space क े space मध ् य space ब िं द ु
space space space space space space space space space space space space space space space space space space left parenthesis straight x comma space straight y right parenthesis space equals space left parenthesis begin inline style fraction numerator 2 space plus space 0 over denominator 2 end fraction end style comma space begin inline style fraction numerator 1 space plus space 3 over denominator space 2 end fraction end style right parenthesis space equals space left parenthesis 1 comma space 2 right parenthesis
straight A space left parenthesis 0 comma space minus 1 right parenthesis comma space straight B space left parenthesis 2 comma space 1 right parenthesis comma space straight C space left parenthesis 0 comma space 3 right parenthesis
increment ABC space क ा space क ् ष े त ् रफल comma
equals space begin inline style 1 half end style left square bracket straight x subscript 1 left parenthesis straight y subscript 2 space minus space straight y subscript 3 right parenthesis space plus space straight x subscript 2 left parenthesis straight y subscript 3 space minus space straight y subscript 1 right parenthesis space plus space straight x subscript 3 left parenthesis straight y subscript 1 space minus space straight y subscript 2 right parenthesis right square bracket
equals space begin inline style 1 half end style left square bracket 0 left parenthesis 1 space minus space 3 right parenthesis space plus space 2 left parenthesis 3 space plus space 1 right parenthesis space plus space 0 left parenthesis negative 1 space minus 1 right parenthesis right square bracket
equals begin inline style 1 half end style left parenthesis 8 right parenthesis space equals space 4 space वर ् ग space य ू न ि ट
straight D left parenthesis 1 comma space 0 right parenthesis comma space straight E left parenthesis 0 comma space 1 right parenthesis space straight F left parenthesis 1 comma space 2 right parenthesis
increment DEF space क ा space क ् ष े त ् रफल comma
equals space begin inline style 1 half end style left square bracket 1 left parenthesis 1 space minus 2 right parenthesis space plus space 0 left parenthesis 2 space minus 0 right parenthesis space plus space 1 left parenthesis 0 space minus space 1 right parenthesis right square bracket space
equals space begin inline style 1 half end style space left parenthesis negative 1 space minus 1 right parenthesis space equals space minus 1
equals space 1 space वर ् ग space य ू न ि ट space space space space space space space space space space space space space space space space space space space space therefore space space space space space क ् ष े त ् रफल space हम े श ा space धन ा त ् मक left parenthesis plus ve right parenthesis space ह ो त ा space ह ै। space

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