document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1034516>Question 1034516</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A square has 12cm sides. Describe its imag for a dilation with center at one of the vertices and scale factor of 0.4. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #649460 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=649460\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if the center is at one of the vertices, then the other vertices on a graph of the square would be:<BR>\n" );
document.write( "(0,0)<BR>\n" );
document.write( "(0,12)<BR>\n" );
document.write( "(12,12)<BR>\n" );
document.write( "(12,0)<BR>\n" );
document.write( "the dilation of .4 would be multiplying all the vertices by .4<BR>\n" );
document.write( ".4 * 12 = 4.8.<BR>\n" );
document.write( "the new coordinates would be:<BR>\n" );
document.write( "(0,0)<BR>\n" );
document.write( "(0,4.8)<BR>\n" );
document.write( "(4.8,4.8)<BR>\n" );
document.write( "(4.8,0)<BR>\n" );
document.write( "the dilation is done by distance from the point of reference.<BR>\n" );
document.write( "since the point of reference is (0,0), then the dilation is done by distance from that point of reference.<BR>\n" );
document.write( "the distance on the graph is d = sqrt((x2-x1)^2 + (y2-y1)^2).<BR>\n" );
document.write( "when x1 and y1 = 0, as in the coordinate (0,0), the formula becomes:<BR>\n" );
document.write( "d = sqrt(x^2 + y^2).<BR>\n" );
document.write( "when x = 0 and y = 12, the formula becomes d = sqrt(0^2 + 12^2) = sqrt(12^2) = 12.<BR>\n" );
document.write( "when you dilate it, you are multiplying the distance by the scale factor which is .4<BR>\n" );
document.write( ".4 * 12 = 4.8.<BR>\n" );
document.write( "the most difficult calculation comes in when x = 12 and y = 12.<BR>\n" );
document.write( "d = sqrt(12^2 + 12^2)<BR>\n" );
document.write( "when you multiply d by .4, you get the new d = .4 * sqrt(12^2 + 12^2).<BR>\n" );
document.write( "if you square d, you get d^2 = .4^2 * sqrt(12^2 + 12^2)^2.<BR>\n" );
document.write( "you can simplify this to get d^2 = .16 * (12^2 + 12^2).<BR>\n" );
document.write( "simplify this further to get d^2 = .16 * 288 = 46.08<BR>\n" );
document.write( "this is the hypotenuse of a right triangle where d is the hypotenuse and x is the adjacent side and y is the opposite side of the angle formed from the center of the graph.<BR>\n" );
document.write( "the right triangle formula for the hypotenuse is d^2 = x^2 + y^2.<BR>\n" );
document.write( "since d^2 = 46.08, and since x = y, the formula becomes 46.08 = 2x^2<BR>\n" );
document.write( "solve for x^2 to get x^2 = 23.04.<BR>\n" );
document.write( "solve for x to get x = sqrt(23.04) = 4.8.<BR>\n" );
document.write( "this means that y = 4.8 as well.<BR>\n" );
document.write( "the coordinate of (12,12) becomes (4.8,4.8).<BR>\n" );
document.write( "if you dilate from the origin, then all dilations are from that point as the reference.<BR>\n" );
document.write( "if you dilate from any other point, than all dilations are from that other point as the reference.<BR>\n" );
document.write( "dilations from the origin are the easiest to perform, but the process would be the same from any other point.<BR>\n" );
document.write( "in that case, the distance formula has to be sqrt((x2-x1)^2 + (y2-y1)^2).\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );