document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=837626>Question 837626</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have a problem:  For a world Peace day celebration the students at cabot high school are making a 6 m by 8 m flag. Each of the six grades will create a motif to honor the people of the six inhabited continets.  Sketch three possible ways to divide the flag. One into six congruent triangles; one into sixs triangles with equal area but none congruent; and one into six congruent trapezoids. Give measuresments or makings on your sketches so each class knows it has equal area </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #504730 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=504730\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> These are the first ideas that came to my mind.<BR>\n" );
document.write( "The possibilities are probably endless, but it is hard enough to draw these three.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Six congruent triangles:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C300%2C-0.4%2C8.4%2C-0.3%2C6.3%2C%0D%0Arectangle%280%2C0%2C8%2C6%29%2Cline%280%2C2%2C8%2C2%29%2C%0D%0Aline%280%2C4%2C8%2C4%29%2Cline%280%2C0%2C8%2C2%29%2C%0D%0Aline%280%2C4%2C8%2C2%29%2Cline%280%2C4%2C8%2C6%29%2Clocate%283.8%2C6%2C8m%29%2C%0D%0Alocate%280.1%2C1.2%2C2m%29%2Clocate%280.1%2C3.2%2C2m%29%2Clocate%280.1%2C5.2%2C2m%29%2C%0D%0Alocate%287.5%2C1.2%2C2m%29%2Clocate%287.5%2C3.2%2C2m%29%2Clocate%287.5%2C5.2%2C2m%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Six right triangles with legs measuring 2m and 8m.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Six congruent trapezoids:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C300%2C-0.4%2C8.4%2C-0.3%2C6.3%2C%0D%0Arectangle%280%2C0%2C8%2C6%29%2Cline%280%2C2%2C8%2C2%29%2C%0D%0Aline%280%2C4%2C8%2C4%29%2Cline%283%2C4%2C5%2C6%29%2C%0D%0Aline%283%2C4%2C5%2C2%29%2Cline%283%2C0%2C5%2C2%29%2C%0D%0Alocate%282.3%2C6%2C5m%29%2Clocate%286.3%2C6%2C3m%29%2C%0D%0Alocate%281.3%2C4%2C3m%29%2Clocate%285.3%2C4%2C5m%29%2C%0D%0Alocate%282.3%2C2%2C5m%29%2Clocate%286.3%2C2%2C3m%29%2C%0D%0Alocate%281.3%2C0.4%2C3m%29%2Clocate%285.3%2C0.4%2C5m%29%2C%0D%0Alocate%280.1%2C1.2%2C2m%29%2Clocate%280.1%2C3.2%2C2m%29%2Clocate%280.1%2C5.2%2C2m%29%2C%0D%0Alocate%287.5%2C1.2%2C2m%29%2Clocate%287.5%2C3.2%2C2m%29%2Clocate%287.5%2C5.2%2C2m%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Six trapezoids with bases measuring 5m and 3m, and a 2m side perpendicular to the bases, for a 2m height.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Six triangles with equal area but none congruent:<BR>\n" );
document.write( "(I would draw this on grid paper. I will draw a grid of 1m by 1m squares).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C300%2C-0.4%2C8.4%2C-0.3%2C6.3%2C%0D%0Arectangle%280%2C0%2C8%2C6%29%2Cline%280%2C2%2C8%2C2%29%2C%0D%0Aline%280%2C4%2C8%2C4%29%2Cline%280%2C1%2C8%2C1%29%2C%0D%0Aline%280%2C3%2C8%2C3%29%2Cline%280%2C5%2C8%2C5%29%2C%0D%0Aline%281%2C0%2C1%2C6%29%2Cline%282%2C0%2C2%2C6%29%2C%0D%0Aline%283%2C0%2C3%2C6%29%2Cline%284%2C0%2C4%2C6%29%2C%0D%0Aline%285%2C0%2C5%2C6%29%2Cline%286%2C0%2C6%2C6%29%2C%0D%0Aline%287%2C0%2C7%2C6%29%2Cred%28line%280%2C6%2C8%2C2%29%29%2C%0D%0Ared%28line%284%2C4%2C8%2C6%29%29%2Cred%28line%288%2C0%2C0%2C6%29%29%2C%0D%0Ared%28line%280%2C2%2C8%2C0%29%29%2Cred%28line%280%2C2%2C4%2C3%29%29%2C%0D%0Alocate%284.1%2C5%2Cred%28A%29%29%2Clocate%283.1%2C0.7%2Cred%28B%29%29%2Clocate%286.1%2C2.5%2Cred%28C%29%29%2C%0D%0Alocate%286.1%2C4%2Cred%28D%29%29%2Clocate%281.3%2C3.7%2Cred%28E%29%29%2Clocate%284.1%2C2%2Cred%28F%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The red lines divide the whole 8m by 6m rectangle (area=(8m)(6m)=48 square meters) into 6 triangles, each with an area of 8 square meters.<BR>\n" );
document.write( "Triangle A is an isosceles triangle with an 8m base and a 2m height, not congruent to any of the other triangles.<BR>\n" );
document.write( "You could say that non-congruent triangles B and C have the same base and height (a 2m base and an 8m height). They are obviously not congruent to any of the other triangles.<BR>\n" );
document.write( "Triangle D is an isosceles triangle with base and height measuring 4 m.<BR>\n" );
document.write( "Triangle E also has a base and a height measuring 4m,but it is not isosceles, and not congruent to any other triangle.<BR>\n" );
document.write( "All those triangles have areas that can be calculated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=base%2Aheight%2F2 ALIGN=MIDDLE  ALT=\"base%2Aheight%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and each has an area of 8 square meters:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2A2%2F2=8 ALIGN=MIDDLE  ALT=\"8%2A2%2F2=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A8%2F2=8 ALIGN=MIDDLE  ALT=\"2%2A8%2F2=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A4%2F2=8 ALIGN=MIDDLE  ALT=\"4%2A4%2F2=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Their areas add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A8 ALIGN=MIDDLE  ALT=\"5%2A8\"   DISABLED_style= \"max-width:90%; height:auto;\" >square meters =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40 ALIGN=MIDDLE  ALT=\"40\"   DISABLED_style= \"max-width:90%; height:auto;\" >square meters.<BR>\n" );
document.write( "The remaining triangle (triangle F) must have an area of<BR>\n" );
document.write( "48 square meters - 40 square meters = 8 square meters.<BR>\n" );
document.write( "If the class assigned triangle F is not convinced enough, you could tell them that E and F together form a triangle with a 4m base and an 8m height, and that E+F triangle is divided into two equal parts by a median, the line segment between triangles E and F.<BR>\n" );
document.write( "If they are still not convinced, they can use Heron's formula to calculate the area, using the lengths of the sides:<BR>\n" );
document.write( "5m (for the side that is half a diagonal of the rectangle), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2817%29 ALIGN=MIDDLE  ALT=\"sqrt%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > meters, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2817%29 ALIGN=MIDDLE  ALT=\"sqrt%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > meters.<BR>\n" );
document.write( "In that case, I would use the version that calculates the area as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F4%29%2Asqrt%284a%5E2b%5E2-%28a%5E2%2Bb%5E2-c%5E2%29%5E2%29 ALIGN=MIDDLE  ALT=\"%281%2F4%29%2Asqrt%284a%5E2b%5E2-%28a%5E2%2Bb%5E2-c%5E2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "(The letters <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the side lengths). </SPAN>\n" );
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