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સમાંતરબાજુ ચતુષ્કોણ અને ત્રિકોણનાં ક્ષેત્રફળ

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ગણિત ધોરણ 9 દ્વિતીય સત્ર

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ગણિત

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bold square to the power of bold m bold space bold ABCD નું ક્ષેત્રફળ 250 સેમી2 છે. જો bold AB with bold bar on top bold space bold અન ે bold space bold CD with bold bar on top નાં મધ્યબિંદુઓ અનુક્રમે E અને F હોય, તો bold square bold space bold AEFD નું ક્ષેત્રફળ શોધો.

bold square to the power of bold m bold space bold ABCDમ ાં bold space bold AB with bold bar on top bold space bold parallel to bold space bold DC with bold bar on top bold space bold તથ ા bold space bold AB bold space bold equals bold space bold DC

bold square to the power of bold m bold space bold ABCD માં E અને F એ અનુક્રમે bold AB with bold bar on top bold space bold અન ે bold space bold CD with bold bar on top નાં મધ્યબિંદુઓ છે.



A – E – B અને D – F – C તથા bold AE bold space bold equals bold space bold 1 over bold 2 bold space bold AB bold space bold અન ે bold space bold DF bold space bold equals bold space bold 1 over bold 2 bold space bold DC થાય.

bold therefore bold space bold square bold space bold AEFDમ ાં bold space bold AE with bold bar on top bold space bold parallel to bold space bold DF with bold bar on top bold space અને AE = DF


આથી bold square bold space bold AEFD સમાંતરબાજુ ચતુષ્કોણ છે.

આ જ રીતે bold square bold space bold BEFC એ પણ સમાંતરબાhસમાંતર બાજુ ચતુષ્કોણ છે.

ધારો કે, bold FM with bold bar on top bold space bold perpendicular bold space bold AB with bold bar on top bold space bold અન ે bold space bold M bold space bold element of bold space bold AB with bold bar on top  છે.

bold square to the power of bold m bold space bold AEFD bold space bold મ ાં bold space bold FM with bold bar on top એ પાયા bold AE with bold bar on top પરનો વેધ છે.


bold therefore bold space bold AEFD bold space bold equals bold space bold FM bold space bold cross times bold space bold AE


આ જ રીતે, 

bold BEFC bold space bold equals bold space bold FM bold space bold cross times bold space bold BE

bold space bold space bold space bold space bold space bold space bold space bold space bold space bold equals bold space bold FM bold space bold cross times bold space bold AE

bold therefore bold space bold AEFD bold space bold equals bold space bold BEFC

bold space bold હવ ે bold comma bold space bold ABCD bold space bold equals bold space bold AEFD bold space bold plus bold space bold BEFC

bold therefore bold space bold 250 bold space bold equals bold space bold AEFD bold space bold plus bold space bold AEFD

bold therefore bold space bold 2 bold space bold AEFD bold space bold equals bold space bold 250

bold therefore bold space bold AEFD bold space bold equals bold space bold 250 over bold 2 bold space bold equals bold space bold 125 bold space bold સ ે મ ી to the power of bold 2


bold square to the power of bold m bold space bold ABCDમ ાં bold space bold CF with bold bar on top bold space bold perpendicular bold space bold AB with bold bar on top bold space bold અન ે bold space bold AE with bold bar on top bold space bold perpendicular bold space bold BC with bold bar on top છે. જો AB = 18 સેમી, AE=  10 સેમી, CF = 12 સેમી, તો AD શોધો.





bold square to the power of bold m bold space bold ABCDમ ાં bold space bold CF with bold bar on top એ પાયા bold AB with bold bar on top પરનો વેધ છે.

bold AE with bold bar on top bold spaceએ પાયા bold BC with bold bar on top પરનો વેધ છે.

bold therefore bold space bold ABCD bold space bold equals bold space bold CF bold space bold cross times bold space bold AB bold space bold equals bold space bold AE bold space bold cross times bold space bold BC

bold therefore bold space bold CF bold space bold cross times bold space bold AB bold space bold equals bold space bold AE bold space bold cross times bold space bold BC

bold therefore bold space bold 12 bold space bold cross times bold space bold 18 bold space bold equals bold space bold 10 bold space bold cross times bold space bold BC

bold therefore bold space bold BC bold space bold equals bold space fraction numerator bold 12 bold space bold cross times bold space bold 18 over denominator bold 10 end fraction

bold therefore bold space bold BC bold space bold equals bold space bold 216 over bold 10 bold space bold equals bold space bold 21 bold. bold 6 bold space bold સ ે મ ી

bold વળ ી bold comma bold space bold square to the power of bold m bold space bold ABCD bold space bold મ ાં bold space bold space bold AD bold space bold equals bold space bold BC

bold therefore bold space bold AD bold space bold equals bold space bold 21 bold. bold 6 bold space bold સ ે મ ી bold space


જો AD = 12 સેમી, CF = 20 સેમી અને AE = 16 સેમી, તો AB શોધો.


bold square to the power of bold m bold space bold ABCDમ ાં bold space bold CF with bold bar on top એ પાયા પરનો bold AB with bold bar on top વેધ છે. તથા bold AE with bold bar on top એ પાયા bold BC with bold bar on top પરનો વેધ છે.


bold therefore bold space bold ABCD bold space bold equals bold space bold CF bold space bold cross times bold space bold AB bold space bold equals bold space bold AE bold space bold cross times bold space bold BC

bold therefore bold space bold CF bold space bold cross times bold space bold AB bold space bold equals bold space bold AE bold space bold cross times bold space bold BC

bold therefore bold space bold 20 bold space bold cross times bold space bold AB bold space bold equals bold space bold 16 bold space bold cross times bold space bold 12

bold therefore bold space bold AB bold space bold equals bold space fraction numerator bold 16 bold space bold cross times bold space bold 12 over denominator bold 20 end fraction

bold therefore bold space bold AB bold space bold equals bold space bold 192 over bold 20 bold space bold equals bold space bold 9 bold. bold 6 bold space bold સ ે મ ી


bold increment bold space bold ABCમાં bold AD with bold bar on top  વેધ છે અને તેને અનુરૂપ પાયો bold BC with bold bar on top છે. વેધ છે અને તેને અનુરૂપ પાયો bold AC with bold bar on top છે. જો AD = 14, BC = 24 અને AC = 35, તો BE શોધો.

bold increment bold space bold ABC માં વેધ bold AD with bold bar on top છે અને તેને અનુરૂપ પાયો bold BC with bold bar on top છે.  bold BE with bold bar on top વેધ છે અને તેને અનુરૂપ પાયો bold AC with bold bar on top છે.



bold therefore bold space bold ABC bold space bold equals bold space bold 1 over bold 2 bold space bold cross times bold space bold AD bold space bold cross times bold space bold BC bold space bold અન ે bold space bold ABC bold space bold equals bold space bold 1 over bold 2 bold space bold cross times bold space bold BE bold space bold cross times bold space bold AC

bold therefore bold space bold 1 over bold 2 bold space bold cross times bold space bold AD bold space bold cross times bold space bold BC bold space bold equals bold space bold 1 over bold 2 bold space bold cross times bold space bold BE bold space bold cross times bold space bold AC

bold therefore bold space bold AD bold space bold cross times bold space bold BC bold space bold equals bold space bold BE bold space bold cross times bold space bold AC

bold therefore bold space bold 14 bold space bold cross times bold space bold 24 bold space bold equals bold space bold BE bold space bold cross times bold space bold 35

bold therefore bold space bold BE bold space bold equals bold space fraction numerator bold 14 bold space bold cross times bold space bold 24 over denominator bold 35 end fraction bold space bold equals bold space bold 48 over bold 4 bold space bold equals bold space bold 96 over bold 10 bold space bold equals bold space bold 9 bold. bold 6


નીચેના વિધાનો ખરા છે કે ખોટા તે જણાવો :
(1) ત્રિકોણ અને ત્રિકોણીય પ્રદેશ બે અલગ ગણ છે.
(2) ત્રિકોણ અને ત્રિકોણના અંદરના ભાગનો છેદગણ ખાલી ગણ છે.
(3) જો ની બાજુઓના મધ્યબિંદુઓ અનુક્રમે D, E અને F હોય, તો bold increment to the power of bold asterisk times bold space bold DEF bold space bold union bold space bold increment to the power of bold asterisk times bold space bold PQR bold space bold equals bold space bold increment to the power of bold asterisk times bold space bold PQR bold.

(4) કોઈ પણ ત્રિકોણ તેના ત્રિકોણીય પ્રદેશનો ઉપગણ છે.
(5) ત્રિકોણનો અંદરનો ભાગ એ ત્રિકોણીય પ્રદેશનો ઉપગણ છે.

 


(1) ત્રિકોણ અને ત્રિકોણીય પ્રદેશ બે અલગ ગણ છે. 

(2) ત્રિકોણ અને ત્રિકોણના અંદરના ભાગનો છેદગણ ખાલી ગણ છે. 

(3) જો ની બાજુઓના મધ્યબિંદુઓ અનુક્રમે D, E અને F હોય, તો bold increment to the power of bold asterisk times bold space bold DEF bold space bold union bold space bold increment to the power of bold asterisk times bold space bold PQR bold space bold equals bold space bold increment to the power of bold asterisk times bold space bold PQR bold. 

(4) કોઈ પણ ત્રિકોણ તેના ત્રિકોણીય પ્રદેશનો ઉપગણ છે. 

(5) ત્રિકોણનો અંદરનો ભાગ એ ત્રિકોણીય પ્રદેશનો ઉપગણ છે.