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ત્રિકોણની બાજુઓની લંબાઇઓ 3 : 4 : 5ના પ્રમાણમાં હોય તથા પરિમિતિ 120 મીટર હોય, તો તે ત્રિકોણનું ક્ષેત્રફળ શોધો.

ધારો કે, ત્રિકોણમાં a : b : c = 3 : 4 : 5 છે અને ત્રિકોણની પરિમિતિ 120 મીટર છે.

ધારો કે, a = 3x હોય, તો b = 4x અને c = 5x

હવે, a + b + c = 120

3x + 4x + 5x = 120

12x = 120

x = 10 મી

હવે, a = 3x = 3 (10) = 30 મી

     b = 4x - 4 (10) = 40 મી

     c = 5x = 5 (10) = 50 મી


હવે, bold s bold space bold equals bold space bold 120 over bold 2 bold space bold equals bold space bold 60 bold space bold મ ી


ત્રિકોણનું ક્ષેત્રફળ bold equals bold space square root of bold s bold space open parentheses bold s bold space bold minus bold space bold a close parentheses bold space open parentheses bold s bold space bold minus bold space bold b close parentheses bold space open parentheses bold s bold space bold minus bold space bold c close parentheses end root

                   bold equals bold space square root of bold 60 bold space open parentheses bold 60 bold space bold minus bold space bold 30 close parentheses bold space open parentheses bold 60 bold space bold minus bold space bold 40 close parentheses bold space open parentheses bold 60 bold space bold minus bold space bold 50 close parentheses end root

bold equals bold space square root of bold 60 bold space bold cross times bold space bold 30 bold space bold cross times bold space bold 20 bold space bold cross times bold space bold 10 end root

bold equals bold space square root of bold 360000

bold equals bold space bold 600 bold space bold મ ી to the power of bold 2


સમદ્વિભુજ ત્રિકોણની પરિમિતિ 30 સેમી અને તેની એકરૂપ બાજુઓ પૈકી પ્રત્યેકની લંબાઈ 12 સેમી હોય, તો તે ત્રિકોણનું ક્ષેત્રફળ શોધો.

bold increment bold space bold ABC માં એકરૂપ બાજુઓ b = c = 12 સેમી અને ત્રિકોણની પરિમિતિ 30 સેમી છે.


bold therefore bold space bold a bold space bold plus bold space bold b bold space bold plus bold space bold c bold space bold equals bold space bold 30

bold therefore bold space bold a bold space bold plus bold space bold 12 bold space bold plus bold space bold 12 bold space bold equals bold space bold 30

bold therefore bold space bold a bold space bold equals bold space bold 30 bold space bold – bold space bold 24 bold space bold equals bold space bold 6 bold space bold સ ે મ ી

bold therefore bold space bold s bold space bold equals bold space bold 30 over bold 2 bold space bold equals bold space bold 15 bold space bold સ ે મ ી



ત્રિકોણનું ક્ષેત્રફળ સેમી bold equals bold space square root of bold s bold space open parentheses bold s bold space bold minus bold space bold a close parentheses bold space open parentheses bold s bold space bold minus bold space bold b close parentheses bold space open parentheses bold s bold space bold minus bold space bold c close parentheses end root


                         bold equals bold space square root of bold 15 bold space open parentheses bold 15 bold space bold minus bold space bold 6 close parentheses bold space open parentheses bold 15 bold space bold minus bold space bold 12 close parentheses bold space open parentheses bold 15 bold space bold minus bold space bold 12 close parentheses end root

bold equals bold space square root of bold 15 bold space bold cross times bold space bold 9 bold space bold cross times bold space bold 3 bold space bold cross times bold space bold 3 end root

bold equals bold space square root of bold 15 bold space bold cross times bold space bold 3 to the power of bold 2 bold space bold cross times bold space bold 3 to the power of bold 2 end root

bold equals bold space bold 9 bold space square root of bold 15 bold space bold સ ે મ ી to the power of bold 2


જે ત્રિકોણની પ્રત્યેક બાજુની લંબાઈ 6 એકમ હોય તેવા સમભુજ ત્રિકોણનું ક્ષેત્રફળ શોધો.


અહીં, પ્રચલિત સંકેતોમાં a = b = c = 6 એકમ

bold therefore bold space bold s bold space bold equals bold space fraction numerator bold a bold space bold plus bold space bold b bold space bold plus bold space bold c over denominator bold 2 end fraction bold space bold equals bold space fraction numerator bold 6 bold space bold plus bold space bold 6 bold space bold plus bold space bold 6 over denominator bold 2 end fraction bold space bold equals bold space bold 18 over bold 2 bold space bold equals bold space bold 9 bold space bold એકમ

સમભુજ ત્રિકોણનું ક્ષેત્રફળ : 

bold equals bold space square root of bold s bold space open parentheses bold s bold space bold minus bold space bold a close parentheses bold space open parentheses bold s bold space bold minus bold space bold b close parentheses bold space open parentheses bold s bold space bold minus bold space bold c close parentheses end root

bold equals bold space square root of bold 9 bold space open parentheses bold 9 bold space bold minus bold space bold 6 close parentheses bold space open parentheses bold 9 bold space bold minus bold space bold 6 close parentheses bold space open parentheses bold 9 bold space bold minus bold space bold 6 close parentheses end root

bold equals bold space square root of bold 9 bold space open parentheses bold 3 close parentheses bold space open parentheses bold 3 close parentheses bold space open parentheses bold 3 close parentheses end root

bold equals bold space square root of bold 9 to the power of bold 2 bold space bold cross times bold space bold 3 end root

bold equals bold space bold 9 bold space square root of bold 3 bold space bold ચ ો bold space bold એકમ


જે ત્રિકોણની બાજુઓની લંબાઈઓ 36 સેમી, 48 સેમી અને 60 સેમી હોય, તે ત્રિકોણનું ક્ષેત્રફળ મેળવો.

અહીં, a = 36 સેમી, b = 48 સેમી અને c = 60 સેમી

bold s bold space bold equals bold space fraction numerator bold a bold space bold plus bold space bold b bold space bold plus bold space bold c over denominator bold 2 end fraction bold space bold equals bold space fraction numerator bold 36 bold space bold plus bold space bold 48 bold space bold plus bold space bold 60 over denominator bold 2 end fraction bold space bold equals bold space bold 144 over bold 2 bold space bold equals bold space bold 72 bold space bold સ ે મ ી

ત્રિકોણનું ક્ષેત્રફળ : 

bold equals bold space square root of bold s bold space open parentheses bold s bold space bold minus bold space bold a close parentheses bold space open parentheses bold s bold space bold minus bold space bold b close parentheses bold space open parentheses bold s bold space bold minus bold space bold c close parentheses end root

bold equals bold space square root of bold 72 bold space open parentheses bold 72 bold space bold minus bold space bold 36 close parentheses bold space open parentheses bold 72 bold space bold minus bold space bold 48 close parentheses bold space open parentheses bold 72 bold space bold minus bold space bold 60 close parentheses end root

bold equals bold space square root of bold 72 bold space bold cross times bold space bold 36 bold space bold cross times bold space bold 24 bold space bold cross times bold space bold 12 end root

bold equals bold space square root of bold 2 bold space bold cross times bold space bold 36 bold space bold cross times bold space bold 36 bold space bold cross times bold space bold 2 bold space bold cross times bold space bold 12 bold space bold cross times bold space bold 12 end root

bold equals bold space square root of bold 2 to the power of bold 2 bold space bold cross times bold space bold 36 to the power of bold 2 bold space bold cross times bold space bold 12 to the power of bold 2 end root

bold equals bold space bold 2 bold space bold cross times bold space bold 36 bold space bold cross times bold space bold 12

bold equals bold space bold 864 bold space bold સ ે મ ી to the power of bold 2


એક કાટકોણ ત્રિકોણના પાયાની લંબાઈ 15 સેમી તથા કર્ણની લંબાઈ 17 સેમી હોય, તો તે ત્રિકોણનું ક્ષેત્રફળ મેળવો.

bold increment bold space bold ABC માં bold m bold space bold angle bold space bold B bold space bold equals bold space bold 90 bold comma સેમી અને BC = 15 સેમી

હવે, પાઈથાગોરસના પ્રમેય મુજબ




bold AC to the power of bold 2 bold space bold equals bold space bold AB to the power of bold 2 bold space bold plus bold space bold BC to the power of bold 2

bold therefore bold space open parentheses bold 17 close parentheses to the power of bold 2 bold space bold equals bold space bold AB to the power of bold 2 bold space bold plus bold space open parentheses bold 15 close parentheses to the power of bold 2

bold therefore bold space bold 289 bold space bold equals bold space bold AB to the power of bold 2 bold space bold plus bold space bold 225

bold therefore bold space bold AB to the power of bold 2 bold space bold equals bold space bold 289 bold space bold minus bold space bold 225 bold space bold equals bold space bold 64

bold therefore bold space bold AB bold space bold equals bold space bold 8



હવે, ત્રિકોણનું ક્ષેત્રફળ = bold 1 over bold 2 bold space bold cross times કાટખૂણો બનાવતી બે બાજુઓનો ગુણાકાર


bold increment bold space bold ABC નું ક્ષેત્રફળ 


bold equals bold space bold 1 over bold 2 bold space bold cross times bold space bold AB bold space bold cross times bold space bold BC


bold equals bold space bold 1 over bold 2 bold space bold cross times bold space bold 8 bold space bold cross times bold space bold 15

bold equals bold space bold 60 bold space bold સ ે મ ી to the power of bold 2