document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=574556>Question 574556</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a 5 x 12 rectangle, a diagonal is drawn and circles are inscribed in both of the right triangles formed. what is the distance between the centers of these circles. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #369207 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=369207\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28360%2C150%2C0%2C36%2C0%2C15%2C%0D%0A%0D%0A%0D%0Aline%280.5%2C0.5%2C35.5%2C0.5%29%2C%0D%0Aline%280.5%2C0.5%2C0.5%2C14.5%29%2C%0D%0Aline%2835.5%2C0.5%2C35.5%2C14.5%29%2C%0D%0Aline%2835.5%2C14.5%2C0.5%2C14.5%29%2C%0D%0Aline%280.5%2C0.5%2C35.5%2C14.5%29%2C%0D%0A%0D%0Acircle%286.2%2C9%2C5.8%29%2C%0D%0Acircle%2836-6.2%2C+15-9%2C+5.8%29%2C%0D%0Aline%286.2%2C9.5%2C6.2%2C14.5%29%2C%0D%0Aline%286.2%2C9.5%2C0.5%2C9.5%29%2C%0D%0Alocate%283.1%2C9.5%2Cr%29%2C%0D%0Alocate%286.4%2C12%2Cr%29%2C%0D%0Aline%286.2%2C9.5%2C+36-6.2%2C+6%29%2C%0D%0A%0D%0Alocate%282%2C2%2CA%29%2C%0D%0Alocate%2834%2C2%2CB%29%2C%0D%0Alocate%2834%2C13%2CC%29%2C%0D%0Alocate%281.5%2C13.5%2CD%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "An easy way to find this distance is to assume that A is the origin of some xy coordinate system. Therefore we may assume that A = (0,0), B = (12,0), C = (12,5), D = (0,5).\r<BR>\n" );
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document.write( "First we should find r. There are several ways to do this, but the easiest way is to use the area formula for triangle ACD:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE Area = rs\">\r<BR>\n" );
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document.write( "where r is the inradius (shown above) and s is the semi-perimeter of ACD ((5+12+13)/2 = 15) Plugging in, we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 30 = 15r \Rightarrow r = 2\">\r<BR>\n" );
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document.write( "Since the two radii meet at right angles with AD and DC, the x-coordinate of the center of the upper-left circle is simply 2. The y-coordinate is 5-2, or 3, so the coordinates of this circle are (2,3).\r<BR>\n" );
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document.write( "Similarly, the coordinates of the center of the lower-right circle are (12-2, 2) or (10,2) (by symmetry). We use the distance formula to find the distance between the two centers:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE d = \sqrt{(10-2)^2 + (2-3)^2} = \sqrt{65}\">. </SPAN>\n" );
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