document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1111119>Question 1111119</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ABCD is a square with side length x cm. M and N are the midpoints of AD and CD respectively. The midpoints and point B make a triangle which is the shaded area. Find the ratio of the shaded to unshaded area. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #726202 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=726202\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> THE PICTURE:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.1%2C1.1%2C-1.1%2C1.1%2C%0D%0Arectangle%28-1%2C-1%2C1%2C1%29%2Clocate%28-0.2%2C0.2%2Cshaded%29%2C%0D%0Atriangle%28-1%2C0%2C0%2C1%2C1%2C-1%29%2Clocate%28-0.2%2C0%2Carea%29%2C%0D%0Alocate%28-1.07%2C-1%2CA%29%2Clocate%281.01%2C-1%2CB%29%2Clocate%281.01%2C1.09%2CC%29%2C%0D%0Alocate%28-1.07%2C1.09%2CD%29%2Clocate%28-1.07%2C0.04%2CM%29%2Clocate%28-0.02%2C1.1%2CN%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "THE FIFTH_GRADER'S SOLUTION:<BR>\n" );
document.write( "If we connect M to the midpoint of BC,<BR>\n" );
document.write( "and N to the midpoint of AB, we split the square into 4 smaller squares, each one being <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > of square ABCD.<BR>\n" );
document.write( "Joining 2 of those small squares we would get a rectangle<BR>\n" );
document.write( "that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > of square ABCD.<BR>\n" );
document.write( "One of those 4 smaller squares<BR>\n" );
document.write( "is divided into two equal triangular parts by diagonal MN,<BR>\n" );
document.write( "so each of those two triangles is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F8 ALIGN=MIDDLE  ALT=\"1%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > of square ABCD.<BR>\n" );
document.write( "Triangle MND is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F8 ALIGN=MIDDLE  ALT=\"1%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > of square ABCD.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.1%2C1.1%2C-1.1%2C1.1%2C%0D%0Arectangle%28-1%2C-1%2C1%2C0%29%2Crectangle%280%2C0%2C1%2C1%29%2C%0D%0Atriangle%28-1%2C0%2C0%2C1%2C-1%2C1%29%2Clocate%28-0.05%2C-0.4%2C1%2F2%29%2C%0D%0Alocate%280.4%2C0.6%2C1%2F4%29%2Clocate%28-0.7%2C0.8%2C1%2F8%29%2Clocate%28-0.4%2C0.4%2C1%2F8%29%2C%0D%0Alocate%28-1.07%2C-1%2CA%29%2Clocate%281.01%2C-1%2CB%29%2Clocate%281.01%2C1.09%2CC%29%2C%0D%0Alocate%28-1.07%2C1.09%2CD%29%2Clocate%28-1.07%2C0.04%2CM%29%2Clocate%28-0.02%2C1.1%2CN%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The diagonal of a rectangle that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4%2B1%2F4=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F4%2B1%2F4=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > of square ABCD<BR>\n" );
document.write( "(such as diagonal MB),<BR>\n" );
document.write( "divides the rectangle into two equal triangles.<BR>\n" );
document.write( "Each of those triangles is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > of square ABCD.<BR>\n" );
document.write( "Triangle ABM is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > of square ABCD, and so is triangle NCB.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.1%2C1.1%2C-1.1%2C1.1%2C%0D%0Atriangle%28-1%2C-1%2C-1%2C0%2C1%2C-1%29%2Ctriangle%281%2C-1%2C-1%2C0%2C1%2C0%29%2C%0D%0Arectangle%280%2C0%2C1%2C1%29%2Clocate%28-0.5%2C-0.5%2C1%2F4%29%2Clocate%280.5%2C-0.3%2C1%2F4%29%2C%0D%0Atriangle%28-1%2C0%2C0%2C1%2C-1%2C1%29%2Clocate%280.4%2C0.6%2C1%2F4%29%2C%0D%0Alocate%28-0.7%2C0.8%2C1%2F8%29%2Clocate%28-0.4%2C0.4%2C1%2F8%29%2C%0D%0Alocate%28-1.07%2C-1%2CA%29%2Clocate%281.01%2C-1%2CB%29%2Clocate%281.01%2C1.09%2CC%29%2C%0D%0Alocate%28-1.07%2C1.09%2CD%29%2Clocate%28-1.07%2C0.04%2CM%29%2Clocate%28-0.02%2C1.1%2CN%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The unshaded area is triangles ABM + NCB + MND, and amounts to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4%2B1%2F4%2B1%2F8=5%2F8 ALIGN=MIDDLE  ALT=\"1%2F4%2B1%2F4%2B1%2F8=5%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the square.<BR>\n" );
document.write( "So, the shaded area is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-5%2F8=3%2F8 ALIGN=MIDDLE  ALT=\"1-5%2F8=3%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the square.<BR>\n" );
document.write( "Obviously the ratio of shaded area to unshaded area is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F8 ALIGN=MIDDLE  ALT=\"3%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%3A%22 ALIGN=MIDDLE  ALT=\"%22%3A%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F8 ALIGN=MIDDLE  ALT=\"5%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%3A5 ALIGN=MIDDLE  ALT=\"3%3A5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "THE EXPECTED SOLUTION is probably as follows<BR>\n" );
document.write( "As ABCD is a square with side length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm\",<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=BC=CD=AD=x ALIGN=MIDDLE  ALT=\"AB=BC=CD=AD=x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm ALIGN=MIDDLE  ALT=\"cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "As \"M and N are the midpoints of AD and CD respectively\",<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AM=MD=DN=NC=x%2F2 ALIGN=MIDDLE  ALT=\"AM=MD=DN=NC=x%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm ALIGN=MIDDLE  ALT=\"cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The unshaded area is made up of triangles <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABM ALIGN=MIDDLE  ALT=\"ABM\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NCB ALIGN=MIDDLE  ALT=\"NCB\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MND ALIGN=MIDDLE  ALT=\"MND\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so its total area is easy to calculate.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28ABM%29 ALIGN=MIDDLE  ALT=\"area%28ABM%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2AAB%2AAM ALIGN=MIDDLE  ALT=\"%281%2F2%29%2AAB%2AAM\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2A%22%28+x%22 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2A%22%28+x%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22cm+%29%22 ALIGN=MIDDLE  ALT=\"%22cm+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22+%22%2A%22+%22 ALIGN=MIDDLE  ALT=\"%22+%22%2A%22+%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28%22 ALIGN=MIDDLE  ALT=\"%22%28%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F2 ALIGN=MIDDLE  ALT=\"x%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22cm+%29%22 ALIGN=MIDDLE  ALT=\"%22cm+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F4 ALIGN=MIDDLE  ALT=\"x%5E2%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28NCB%29 ALIGN=MIDDLE  ALT=\"area%28NCB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2ACB%2ANC ALIGN=MIDDLE  ALT=\"%281%2F2%29%2ACB%2ANC\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2A%22%28+x%22 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2A%22%28+x%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22cm+%29%22 ALIGN=MIDDLE  ALT=\"%22cm+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22+%22%2A%22+%22 ALIGN=MIDDLE  ALT=\"%22+%22%2A%22+%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28%22 ALIGN=MIDDLE  ALT=\"%22%28%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F2 ALIGN=MIDDLE  ALT=\"x%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22cm+%29%22 ALIGN=MIDDLE  ALT=\"%22cm+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F4 ALIGN=MIDDLE  ALT=\"x%5E2%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28MND%29 ALIGN=MIDDLE  ALT=\"area%28MND%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2AMD%2AND ALIGN=MIDDLE  ALT=\"%281%2F2%29%2AMD%2AND\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2A%22%28%22 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2A%22%28%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F2 ALIGN=MIDDLE  ALT=\"x%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22cm+%29%22 ALIGN=MIDDLE  ALT=\"%22cm+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22+%22%2A%22+%22 ALIGN=MIDDLE  ALT=\"%22+%22%2A%22+%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28%22 ALIGN=MIDDLE  ALT=\"%22%28%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F2 ALIGN=MIDDLE  ALT=\"x%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22cm+%29%22 ALIGN=MIDDLE  ALT=\"%22cm+%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F8 ALIGN=MIDDLE  ALT=\"x%5E2%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=unshaded ALIGN=MIDDLE  ALT=\"unshaded\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area ALIGN=MIDDLE  ALT=\"area\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2F4%2Bx%5E2%2F4%2Bx%5E2%2F8%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2F4%2Bx%5E2%2F4%2Bx%5E2%2F8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2F8%29x%5E2 ALIGN=MIDDLE  ALT=\"%285%2F8%29x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Calculating the shaded area as the area of triangle MNB may seem difficult,<BR>\n" );
document.write( "However,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28MNB%29 ALIGN=MIDDLE  ALT=\"area%28MNB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=shaded ALIGN=MIDDLE  ALT=\"shaded\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area ALIGN=MIDDLE  ALT=\"area\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28ABCD%29 ALIGN=MIDDLE  ALT=\"area%28ABCD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22-%22 ALIGN=MIDDLE  ALT=\"%22-%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=unshaded ALIGN=MIDDLE  ALT=\"unshaded\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area ALIGN=MIDDLE  ALT=\"area\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28ABCD%29 ALIGN=MIDDLE  ALT=\"area%28ABCD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB%2ABC ALIGN=MIDDLE  ALT=\"AB%2ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28+x%22 ALIGN=MIDDLE  ALT=\"%22%28+x%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22cm+%29%22%5E2 ALIGN=MIDDLE  ALT=\"%22cm+%29%22%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28MNB%29 ALIGN=MIDDLE  ALT=\"area%28MNB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=shaded ALIGN=MIDDLE  ALT=\"shaded\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area ALIGN=MIDDLE  ALT=\"area\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22-%22 ALIGN=MIDDLE  ALT=\"%22-%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2F8%29x%5E2 ALIGN=MIDDLE  ALT=\"%285%2F8%29x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F8%29x%5E2 ALIGN=MIDDLE  ALT=\"%283%2F8%29x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The ratio of those two area values, both in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%5E2 ALIGN=MIDDLE  ALT=\"cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%283%2F8%29%29%2F%28%285%2F8%29%29 ALIGN=MIDDLE  ALT=\"%28%283%2F8%29%29%2F%28%285%2F8%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F8%29%2A%288%2F5%29 ALIGN=MIDDLE  ALT=\"%283%2F8%29%2A%288%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%283%2F5%29 ALIGN=MIDDLE  ALT=\"highlight%283%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%283%3A5%29 ALIGN=MIDDLE  ALT=\"highlight%283%3A5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );