document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190310>Question 1190310</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two poles have wires running from the top of each to the base of the base of the other. If the poles are 8 m and 12 m high, at what height do the wires cross? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #821931 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=821931\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let's say we had these four points<BR>\n" );
document.write( "A = (0,0)<BR>\n" );
document.write( "B = (0,8)<BR>\n" );
document.write( "C = (p,0)<BR>\n" );
document.write( "D = (p,12)<BR>\n" );
document.write( "I'll make p > 0, but it easily works for negative values of p as well.<BR>\n" );
document.write( "The only condition really is that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+%3C%3E+0 ALIGN=MIDDLE  ALT=\"p+%3C%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Joining points A and B gets segment AB which is the first tower (8 meters)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Joining points C and D gets segment CD which is the second tower (12 meters)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The distance between the base of each tower (A and C) is exactly p units. <BR>\n" );
document.write( "It will turn out that the value of p doesn't matter.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's find the equation of line BC<BR>\n" );
document.write( "m = slope<BR>\n" );
document.write( "m = (y2-y1)/(x2-x1)<BR>\n" );
document.write( "m = (0-8)/(p-0)<BR>\n" );
document.write( "m = -8/p<BR>\n" );
document.write( "The y intercept of line BC is b = 8 because of point B on the y axis.<BR>\n" );
document.write( "y = mx+b<BR>\n" );
document.write( "y = (-8/p)x+8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The equation of line BC is: y = (-8/p)x+8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Through similar steps, the equation of line AD is y = (12/p)x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let point E be the intersection of those two lines. The goal is to find the y coordinate of point E.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Apply substitution to get<BR>\n" );
document.write( "y = (-8/p)x+8<BR>\n" );
document.write( "(12/p)x = (-8/p)x+8<BR>\n" );
document.write( "12x = -8x + 8p<BR>\n" );
document.write( "12x+8x = 8p<BR>\n" );
document.write( "20x = 8p<BR>\n" );
document.write( "x = 8p/20<BR>\n" );
document.write( "x = 2p/5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then plug this into either equation for BC or AD<BR>\n" );
document.write( "y = (-8/p)*x + 8<BR>\n" );
document.write( "y = (-8/p)*(2p/5) + 8<BR>\n" );
document.write( "y = -16/5 + 8<BR>\n" );
document.write( "y = -16/5 + 40/5<BR>\n" );
document.write( "y = (-16 + 40)/5<BR>\n" );
document.write( "y = 24/5<BR>\n" );
document.write( "y = 4.8<BR>\n" );
document.write( "Or<BR>\n" );
document.write( "y = (12/p)*x<BR>\n" );
document.write( "y = (12/p)*(2p/5)<BR>\n" );
document.write( "y = 24/5<BR>\n" );
document.write( "y = 4.8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Either way we get y = 4.8 as the height of the intersection of the two lines.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In either case, the p variables cancel out. <BR>\n" );
document.write( "This means that the horizontal distance between the poles doesn't matter.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: <font color=red>4.8 meters</font> (this value is exact).<BR>\n" );
document.write( "</font><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );