document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=267085>Question 267085</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The length of the hypotenuse of a right-angled triangle is 240 units. The perimeter of the given triangle is a perfect square. If the perimeter of the given triangle is greater than 550 units, then which of the following can be the length of a side of the given right-angled triangle?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #196120 by </B><A HREF=/tutors/aboutme.mpl?userid=CharlesG2><B>CharlesG2(834)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CharlesG2><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=196120\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The length of the hypotenuse of a right-angled triangle is 240 units. The perimeter of the given triangle is a perfect square. If the perimeter of the given triangle is greater than 550 units, then which of the following can be the length of a side of the given right-angled triangle? \r<BR>\n" );
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document.write( "a^2 + b^2 = c^2 = 240^2 = 57600<BR>\n" );
document.write( "a + b + 240 > 550<BR>\n" );
document.write( "a + b > 310<BR>\n" );
document.write( "a > 310 - b<BR>\n" );
document.write( "b > 310 - a<BR>\n" );
document.write( "first perfect square after 550 is 24^2 which is 576<BR>\n" );
document.write( "set a + b + 240 equal to 576<BR>\n" );
document.write( "a + b = 336 (576-240=336)<BR>\n" );
document.write( "a = 336 - b<BR>\n" );
document.write( "b = 336 - a (use this to subsitute)<BR>\n" );
document.write( "a^2 + (336 - a)^2 = 57600<BR>\n" );
document.write( "a^2 + 112896 - 672a + a^2 = 57600 (336^2=112896, 2*336=672)<BR>\n" );
document.write( "2a^2 - 672a + 55296 = 0<BR>\n" );
document.write( "a^2 - 336a + 27648 = 0<BR>\n" );
document.write( "27648 = 2 * 13824<BR>\n" );
document.write( "13824 = 2 * 6912<BR>\n" );
document.write( "6912 = 2 * 3456<BR>\n" );
document.write( "3456 = 2 * 1728<BR>\n" );
document.write( "1728 = 2 * 864<BR>\n" );
document.write( "864 = 2 * 432<BR>\n" );
document.write( "432 = 2 * 216<BR>\n" );
document.write( "216 = 2 * 108<BR>\n" );
document.write( "108 = 2 * 54<BR>\n" );
document.write( "54 = 2 * 27<BR>\n" );
document.write( "27 = 3 * 3 * 3<BR>\n" );
document.write( "27648 = 2^10 * 3^3 = 1024 * 27<BR>\n" );
document.write( "144 = 2^4 * 3^2<BR>\n" );
document.write( "27648 = 2^4 * 3^2 * 2^6 * 3 = 16 * 9 * 64 * 3 = 144 * 192<BR>\n" );
document.write( "144 + 192 = 336<BR>\n" );
document.write( "(a - 192)(a - 144)=0<BR>\n" );
document.write( "a = 144 or a = 192<BR>\n" );
document.write( "now b = 336 - a<BR>\n" );
document.write( "so b = 192 or b = 144<BR>\n" );
document.write( "side a needs to be a minimum of 144, and side b needs to be a minimum of 192\r<BR>\n" );
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document.write( " </SPAN>\n" );
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