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Tentukan koordinat titik-titik oleh rotasi R denga...

Tentukan koordinat titik-titik oleh rotasi R dengan sudut begin mathsize 14px style alpha end style dan pusat P serta arah rotasi sebagai berikut:

Jawaban:

Rotasi

Jika suatu titik begin mathsize 14px style P left parenthesis x comma y right parenthesis end style diputar sejauh begin mathsize 14px style alpha end style berlawanan dengan arah jam terhadap titik pusat begin mathsize 14px style A left parenthesis a comma b right parenthesis end style maka bayangannya adalah

begin mathsize 14px style P ’ left parenthesis x ’ comma space y ’ right parenthesis end style dengan :

begin mathsize 14px style open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row cell cos space alpha end cell cell negative sin space alpha end cell row cell sin space alpha end cell cell cos space alpha end cell end table close parentheses open parentheses table row cell x minus a end cell row cell x minus b end cell end table close parentheses plus open parentheses table row a row b end table close parentheses end style 

Jika suatu titik begin mathsize 14px style P left parenthesis x comma y right parenthesis end style diputar sejauh begin mathsize 14px style alpha end style searah dengan arah jam terhadap titik pusat begin mathsize 14px style A left parenthesis a comma b right parenthesis end style maka bayangannya adalah

begin mathsize 14px style P ’ left parenthesis x ’ comma space y ’ right parenthesis end style dengan :

begin mathsize 14px style open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row cell cos space alpha end cell cell sin space alpha end cell row cell negative sin space alpha end cell cell cos space alpha end cell end table close parentheses open parentheses table row cell x minus a end cell row cell x minus b end cell end table close parentheses plus open parentheses table row a row b end table close parentheses end style 

Secara umum diperoleh rumus berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x apostrophe minus a end cell equals cell open parentheses x minus a close parentheses cos space alpha minus open parentheses y minus b close parentheses sin space alpha end cell row cell y apostrophe minus b end cell equals cell open parentheses x minus a close parentheses sin space alpha plus open parentheses y minus b close parentheses cos space alpha end cell end table end style 

Untuk begin mathsize 14px style E open parentheses 2 comma 2 close parentheses end style diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x apostrophe plus 1 end cell equals cell open parentheses 2 plus 1 close parentheses cos space open parentheses negative 45 degree close parentheses minus open parentheses 2 plus 2 close parentheses sin space open parentheses negative 45 degree close parentheses end cell row cell x apostrophe plus 1 end cell equals cell 3 times cos space left parenthesis 45 degree right parenthesis plus 4 times sin space left parenthesis 45 degree right parenthesis end cell row cell x apostrophe plus 1 end cell equals cell 3 times 1 half square root of 2 plus 4 times 1 half square root of 2 end cell row cell x apostrophe end cell equals cell 7 over 2 square root of 2 minus 1 end cell row cell y apostrophe plus 2 end cell equals cell open parentheses 2 plus 1 close parentheses sin space open parentheses negative 45 degree close parentheses plus open parentheses 2 plus 2 close parentheses cos left parenthesis negative 45 degree right parenthesis end cell row cell y apostrophe plus 2 end cell equals cell negative 3 times sin space left parenthesis 45 degree right parenthesis plus 4 times cos left parenthesis 45 degree right parenthesis end cell row cell y apostrophe plus 2 end cell equals cell negative 3 times 1 half square root of 2 plus 4 times 1 half square root of 2 end cell row cell y apostrophe end cell equals cell 1 half square root of 2 minus 2 end cell end table end style    

Jadi, begin mathsize 14px style E apostrophe open parentheses 7 over 2 square root of 2 minus sign 1 comma 1 half square root of 2 minus sign 2 close parentheses end style.

 

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