document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191633>Question 1191633</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the image of point  (5, 3)  when it is rotated  1/4 of a turn counter-clockwise around the point  (-1,4) .  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823418 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=823418\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The center of rotation is (-1,4)<BR>\n" );
document.write( "We need to translate this center of rotation to the origin (0,0)\r<BR>\n" );
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document.write( "Apply the translation rule <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large (x,y) \to (x+1,y-4)\"> so that (-1,4) moves to (0,0)<BR>\n" );
document.write( "This will move (5,3) to (6,-1)\r<BR>\n" );
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document.write( "Another way to look at it: imagine that the xy axis is allowed to move but the points (5,3) and (-1,4) stay fixed.<BR>\n" );
document.write( "If you moved the xy axis to have (-1,4) placed at the intersection of the axes, then it will give the illuison of movement described earlier.\r<BR>\n" );
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document.write( "Now that we have the center of rotation be the origin, we apply this 90 degree counter-clockwise rule:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large (x,y) \to (-y,x)\"><BR>\n" );
document.write( "This means (6,-1) becomes (1,6) after the counter-clockwise quarter turn.\r<BR>\n" );
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document.write( "After the rotation is done, we have to undo the translation we initially did. This will get us back to the correct frame of reference so to speak.\r<BR>\n" );
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document.write( "The translation we did was<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large (x,y) \to (x+1,y-4)\"><BR>\n" );
document.write( "the inverse or opposite translation would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large (x,y) \to (x-1,y+4)\"><BR>\n" );
document.write( "That moves the point (1,6) to (0,10) which is the final answer.\r<BR>\n" );
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document.write( "Visual Confirmation (I used GeoGebra to make the graph):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/4XTJS9C.png\">\r<BR>\n" );
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document.write( "Answer: (0,10)<BR>\n" );
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