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સમરૂપતા અને પાયથાગોરસ પ્રમેય

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CBSE Gujarat Board Haryana Board

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સમરૂપતા અને પાયથાગોરસ પ્રમેય
bold increment bold space bold ABCમાં  bold angle bold Bકાટકોણ છે. top enclose bold BD bold perpendicular bold space top enclose bold AC અને bold D bold space bold element of bold space top enclose bold AC bold space bold. જો bold AD bold space bold equals bold space bold 4 bold DC તો સાબિત કરો કે bold BD bold space bold equals bold space bold 2 bold DC bold. 
Answer
    • એક ત્રિકોણની બાજુઓનાં માપ 5, 12 અને 13 છે. સાબિત કરો કે ત્રિકોણ કાટકોણ ત્રિકોણ છે. ત્રિકોણના કર્ણ પરના વેધની લંબાઇ શોધો.
    Answer
    • bold increment bold space bold PQR માં top enclose bold QM એ કર્ણ top enclose bold PR પરનો વેધ છે. જો bold PM bold space bold equals bold space bold 8 bold comma bold space bold RM bold space bold equals bold space bold 12 bold space હોય, તો bold PQ bold comma bold space bold QR અને bold QM શોધો. 
    Answer
    • bold increment bold space bold ABC bold spaceમાં bold m bold space bold angle bold space bold B bold space bold equals bold space bold 90 bold degree bold comma bold space top enclose bold BM bold space bold perpendicular bold space top enclose bold AC bold comma bold space bold M bold space bold element of bold space top enclose bold AC bold space bold. bold spaceજો bold AM bold space bold minus bold space bold MC bold space bold space bold equals bold space bold 7 bold comma bold space bold AB to the power of bold 2 bold space bold minus bold space bold AC to the power of bold 2 bold space bold equals bold space bold 175 bold space હોય, તો AC શોધો. 
    Answer
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    bold square bold space bold PQRS bold space લંબચોરસ છે.  જો bold PQ bold space bold plus bold space bold QR bold space bold equals bold space bold 7 bold space અને bold PR bold space bold plus bold space bold QS bold space bold equals bold space bold 10 bold space હોય, તો bold square bold space bold PQRS નું ક્ષેત્રફળ શોધો. 



    લંબચોરસ bold square bold space bold PQRS bold space ના વિકર્ણો એકરૂપ હોય.

    bold therefore bold space bold PR bold space bold equals bold space bold QS bold space 

    પરંતુ bold PR bold space bold plus bold space bold QS bold space bold equals bold space bold 10 bold space 

    bold therefore bold PR bold space bold equals bold space bold QS bold space bold equals bold space bold 5 bold space

    bold PQ bold space bold plus bold space bold QR bold space bold equals bold space bold 7 bold spaceમાં bold PQ bold space bold equals bold space bold x ધારો

    bold therefore bold space bold QR bold space bold equals bold space bold 7 bold space bold space bold minus bold space bold x

    bold increment bold space bold PQR માં bold m bold space bold angle bold space bold Q bold space bold equals bold space bold 90 bold degree bold space

     
    bold therefore bold space bold PQ to the power of bold 2 bold space bold plus bold space bold QR to the power of bold 2 bold space bold equals bold space bold PR to the power of bold 2 bold space bold therefore bold space bold x to the power of bold 2 bold space bold plus bold space bold left parenthesis bold 7 bold space bold minus bold space bold x bold right parenthesis to the power of bold 2 bold space bold equals bold space bold 5 to the power of bold 2 bold space bold therefore bold space bold x to the power of bold 2 bold space bold plus bold space bold 49 bold space bold minus bold space bold 14 bold x bold space bold plus bold space bold x to the power of bold 2 bold space bold equals bold space bold 25 bold space bold therefore bold space bold 2 bold x to the power of bold 2 bold space bold minus bold space bold 14 bold x bold space bold plus bold space bold 49 bold space bold minus bold space bold 25 bold space bold equals bold space bold 0 bold space bold therefore bold space bold 2 bold x to the power of bold 2 bold space bold minus bold space bold 14 bold x bold space bold plus bold space bold space bold 24 bold space bold equals bold space bold 0 bold space bold therefore bold space bold x to the power of bold 2 bold space bold minus bold space bold 7 bold X bold space bold plus bold space bold 12 bold space bold equals bold space bold 0 bold space bold space bold space bold space bold space bold left parenthesis bold space bold because bold space bold 2 bold space bold not equal to bold 0 bold right parenthesis bold space bold therefore bold space bold x to the power of bold 2 bold space bold minus bold space bold 4 bold x bold space bold minus bold space bold 3 bold x bold space bold plus bold space bold 12 bold space bold equals bold space bold 0 bold space bold therefore bold space bold x bold space bold left parenthesis bold x bold space bold minus bold space bold 4 bold right parenthesis bold space bold times bold space bold 3 bold space bold left parenthesis bold x bold space bold minus bold space bold 4 bold right parenthesis bold space bold equals bold space bold 0 bold space bold therefore bold space bold x bold space bold left parenthesis bold x bold space bold minus bold space bold 4 bold right parenthesis bold space bold times bold space bold 3 bold space bold left parenthesis bold x bold space bold minus bold space bold 4 bold right parenthesis bold space bold equals bold space bold 0 bold space bold therefore bold left parenthesis bold x bold space bold minus bold space bold 4 bold right parenthesis bold space bold left parenthesis bold x bold space bold minus bold space bold 3 bold right parenthesis bold space bold equals bold space bold 0 bold space bold therefore bold space bold x bold space bold equals bold space bold 4 bold space bold અથવ ા bold space bold space bold x bold space bold equals bold space bold 3 bold space


    જો bold PQ bold space bold equals bold space bold 4 bold space હોય, તો bold QR bold space bold equals bold space bold 7 bold space bold minus bold space bold 4 bold space bold equals bold space bold 3 bold space થાય.

    જો bold PQ bold space bold equals bold space bold 3 હોય, તો bold QR bold space bold equals bold space bold 7 bold space bold minus bold space bold 3 bold space bold equals bold space bold 4 bold space થાય.

    bold therefore bold space bold square bold space bold PQRS નું ક્ષેત્રફળ  bold equals bold space bold PQ bold space bold cross times bold space bold QR bold space

                                   bold equals bold space bold 3 bold space bold cross times bold space bold 4 bold space bold equals bold space bold 12 bold spaceથાય.

    Answer
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