चतुर्भुजों को समझना

Question

निम्नलिखित प्रत्येक में कितने विकर्ण हैं?

(a) एक उत्तल चतुर्भुज (b) एक समषड्भुज (c) एक त्रिभुज

Solution

straight n minus भ ु ज ा ओ ं space क े space बह ु भ ु ज space म ें space व ि कर ् ण ों space क ी space स ं ख ् य ा space equals space open square brackets space fraction numerator straight n space left parenthesis space straight n space minus space 1 space right parenthesis over denominator 2 end fraction space minus space straight n space close square brackets

(a) एक उत्तल चतुर्भुज में भुजाओं की संख्या (n) = 4

व ि कर ् ण ों space क ी space स ं ख ् य ा space equals space fraction numerator straight n space left parenthesis space straight n space minus space 1 space right parenthesis over denominator 2 end fraction space minus space straight n equals space fraction numerator 4 space left parenthesis space 4 space minus space 1 space right parenthesis over denominator 2 end fraction space minus space 4 space equals space fraction numerator 4 space straight x space 3 over denominator 2 end fraction space minus space 4 equals space 12 over 2 space minus space 4 space equals space 6 space minus space 4 space equals space 2

(b) एक समषड्भुज में भिजाओं की संख्या (n) = 6

व ि कर ् ण ों space क ी space स ं ख ् य ा space equals space open square brackets space fraction numerator straight n space left parenthesis space straight n space minus space 1 space right parenthesis over denominator 2 end fraction space minus space straight n space close square brackets equals space open square brackets space fraction numerator 6 space left parenthesis space 6 space minus space 1 space right parenthesis over denominator 2 end fraction space minus space 6 space close square brackets space equals space fraction numerator 6 space straight x space 5 over denominator 2 end fraction space minus space 6 equals space 15 space minus space 6 space equals space 9

व ि कर ् ण ों space क ी space स ं ख ् य ा space equals space open square brackets space fraction numerator straight n space left parenthesis space straight n space minus space 1 space right parenthesis over denominator 2 end fraction space minus space straight n space close square brackets equals space fraction numerator 3 space left parenthesis space 3 space minus space 1 space right parenthesis over denominator 2 end fraction space minus space 3 space equals space fraction numerator 3 space straight x space 2 over denominator 2 end fraction space minus space 3 equals space 3 space minus space 3 space equals space 0