document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1070529>Question 1070529</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let ABC be a triangle such that AB= 6cm, AC=8cm and BC 10cm and M be the midpoint<BR>\n" );
document.write( "of BC .AMDE is a square, MD and intersects AC at point F . Find the area of quadrilateral AFDE. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #685594 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=685594\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here is triangle ABC with midsegments LM and MN <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28360%2C300%2C-5%2C7%2C-1%2C9%2C%0D%0Agreen%28rectangle%280%2C0%2C3%2C4%29%29%2C%0D%0Ared%28triangle%280%2C0%2C6%2C0%2C0%2C8%29%29%2C%0D%0Aline%280%2C0%2C3%2C4%29%2C%0D%0Alocate%283%2C4.5%2CM%29%2Clocate%286%2C0%2CB%29%2C%0D%0Alocate%28-0.2%2C0%2CA%29%2Clocate%28-0.1%2C8.5%2CC%29%2C%0D%0Alocate%28-0.3%2C4.2%2CL%29%2Clocate%282.9%2C0%2CN%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You know that those midsegments are parallel to<BR>\n" );
document.write( "and half the length of the side they do not touch.<BR>\n" );
document.write( "That makes ALMN a rectangle.<BR>\n" );
document.write( "Line AM is the diagonal of rectangle ALMN.<BR>\n" );
document.write( "Triangle ABC is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%283%29 ALIGN=MIDDLE  ALT=\"red%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > times scaled-up version of the most popular 3-4-5 right triangle,<BR>\n" );
document.write( "with side lengths in cm of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=3%2Ared%282%29=6 ALIGN=MIDDLE  ALT=\"AB=3%2Ared%282%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC=4%2Ared%282%29=8 ALIGN=MIDDLE  ALT=\"AC=4%2Ared%282%29=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=5%2Ared%282%29=10 ALIGN=MIDDLE  ALT=\"BC=5%2Ared%282%29=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It has a smaller acute angle ACB (opposite shorter side AB),<BR>\n" );
document.write( "a larger acute angle ABC, and a right angle BAC (opposite longest side BC).<BR>\n" );
document.write( "The mid segments, and AM split triangle ABC into four smaller triangles.<BR>\n" );
document.write( "They are all 3-4-5 right triangles with sides measuring<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=LM=AN=NB=3cm ALIGN=MIDDLE  ALT=\"LM=AN=NB=3cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AL=LC=CM=4cm ALIGN=MIDDLE  ALT=\"AL=LC=CM=4cm\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BM=MC=highlight%28AM=5cm%29 ALIGN=MIDDLE  ALT=\"BM=MC=highlight%28AM=5cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "congruent smaller acute angles (so that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CAM=ACM ALIGN=MIDDLE  ALT=\"CAM=ACM\"   DISABLED_style= \"max-width:90%; height:auto;\" > for example),<BR>\n" );
document.write( "and congruent larger acute angles, as well.<BR>\n" );
document.write( "Since they tell us that \"MD and intersects AC at point F\",<BR>\n" );
document.write( "I know to what side of segment AM square AMDE is locate,<BR>\n" );
document.write( "and I can draw the square, like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28360%2C300%2C-5%2C7%2C-1%2C9%2C%0D%0Agreen%28rectangle%280%2C0%2C3%2C4%29%29%2C%0D%0Ared%28triangle%280%2C0%2C6%2C0%2C0%2C8%29%29%2C%0D%0Aline%280%2C0%2C3%2C4%29%2Cline%28-4%2C3%2C-1%2C7%29%2C%0D%0Aline%280%2C0%2C-4%2C3%29%2Cline%283%2C4%2C-1%2C7%29%2C%0D%0Alocate%283%2C4.5%2CM%29%2Clocate%286%2C0%2CB%29%2C%0D%0Alocate%28-0.2%2C0%2CA%29%2Clocate%28-0.1%2C8.5%2CC%29%2C%0D%0Alocate%28-4.2%2C3%2CE%29%2Clocate%28-1.1%2C7.5%2CD%29%2C%0D%0Alocate%28-0.3%2C6.3%2CF%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28AFDE%29=area%28AMDE%29-area%28MFA%29 ALIGN=MIDDLE  ALT=\"area%28AFDE%29=area%28AMDE%29-area%28MFA%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28AMDE%29=AM%5E2=%285cm%29%5E2=25cm%5E2 ALIGN=MIDDLE  ALT=\"area%28AMDE%29=AM%5E2=%285cm%29%5E2=25cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and triangle MFA is a right triangle,<BR>\n" );
document.write( "similar to LMA and all the other 3-4-5 triangles we looked at before,<BR>\n" );
document.write( "because it shares the same smaller acute angle CAM with triangle LMA.<BR>\n" );
document.write( "So, for MFA, the shorter leg to longer leg ratio is the same<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MF%2FAM=3%2F4 ALIGN=MIDDLE  ALT=\"MF%2FAM=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MF%2F%223+cm%22=3%2F4 ALIGN=MIDDLE  ALT=\"MF%2F%223+cm%22=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MF=%283%2F4%29%283cm%29 ALIGN=MIDDLE  ALT=\"MF=%283%2F4%29%283cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MF=9%2F4 ALIGN=MIDDLE  ALT=\"MF=9%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cm%29 ALIGN=MIDDLE  ALT=\"cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28MFA%29=%281%2F2%29%2AAM%2AMF ALIGN=MIDDLE  ALT=\"area%28MFA%29=%281%2F2%29%2AAM%2AMF\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28MFA%29=%281%2F2%29%2A3%2A%289%2F4%29 ALIGN=MIDDLE  ALT=\"area%28MFA%29=%281%2F2%29%2A3%2A%289%2F4%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;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28MFA%29=27%2F8%7D%7D%7B%7B%7Bcm%5E2 ALIGN=MIDDLE  ALT=\"area%28MFA%29=27%2F8%7D%7D%7B%7B%7Bcm%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%28AFDE%29=area%28AMDE%29-area%28MFA%29 ALIGN=MIDDLE  ALT=\"area%28AFDE%29=area%28AMDE%29-area%28MFA%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area%28AMDE%29=25cm%5E2-3.375cm%5E2=highlight%2821.625cm%5E2%29 ALIGN=MIDDLE  ALT=\"area%28AMDE%29=25cm%5E2-3.375cm%5E2=highlight%2821.625cm%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );