document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208980>Question 1208980</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the diagram AD = 4 cm, EF = 3 cm, and parallel segments are indicated. If the total area of the trapezoid is 135 cm2, what is the area, in cm2, of triangle AMN?<BR>\n" );
document.write( "https://ibb.co/ggBgSQB </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847535 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=847535\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll recreate the diagram just in case the link might expire (at some point in the future).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-1%2C12%2C-1%2C19%2C%0D%0Aline%282.46%2C18%2C0%2C0%29%2Cline%280%2C0%2C11%2C0%29%2Cline%2811%2C0%2C6.46%2C18%29%2Cline%286.46%2C18%2C2.46%2C18%29%2Cline%282.46%2C18%2C7%2C0%29%2Cline%286.46%2C18%2C4%2C0%29%2Cline%282.46%2C18%2C11%2C0%29%2C%0D%0A%0D%0Aline%282.46%2C18%2C1.66190558%2C12.160284732%29%2Cline%281.66190558%2C12.160284732%2C1.43900558%2C12.705884732%29%2Cline%281.66190558%2C12.160284732%2C2.02300558%2C12.626084732%29%2Cline%282.46%2C18%2C4.87%2C18%29%2Cline%284.87%2C18%2C4.6613%2C17.8795%29%2Cline%284.87%2C18%2C4.6613%2C18.1205%29%2Cline%282.46%2C18%2C4.0969258628%2C11.5099855662%29%2Cline%284.0969258628%2C11.5099855662%2C3.6306258628%2C11.9901855662%29%2Cline%284.0969258628%2C11.5099855662%2C4.2796258628%2C12.1538855662%29%2Cline%286.46%2C18%2C5.8493574143%2C13.531883519%29%2Cline%285.8493574143%2C13.531883519%2C5.6788574143%2C13.949383519%29%2Cline%285.8493574143%2C13.531883519%2C6.1256574143%2C13.888283519%29%2Cline%286.46%2C18%2C8.1243641745%2C11.4011993081%29%2Cline%288.1243641745%2C11.4011993081%2C7.6502641745%2C11.8894993081%29%2Cline%288.1243641745%2C11.4011993081%2C8.3101641745%2C12.0558993081%29%2Cline%287%2C0%2C9.19%2C0%29%2Cline%289.19%2C0%2C9.0003%2C-0.1095%29%2Cline%289.19%2C0%2C9.0003%2C0.1095%29%2C%0D%0A%0D%0Alocate%282.46-0.5%2C18%2C%22A%22%29%2Clocate%280%2C0%2C%22B%22%29%2Clocate%2811%2C0%2C%22C%22%29%2Clocate%286.46%2B0.5%2C18%2C%22D%22%29%2Clocate%284%2C0%2C%22E%22%29%2Clocate%287%2C0%2C%22F%22%29%2Clocate%285.5654545455%2B0.5%2C11.4545454545%2C%22M%22%29%2Clocate%285.0542857143%2B0.5%2C7.7142857143%2C%22N%22%29%2C%0D%0Alocate%285.2%2C0%2C%223+cm%22%29%2Clocate%284.2%2C18%2C%224+cm%22%29%2C%0D%0Alocate%289%2C18%2Cmatrix%281%2C2%2C%22Diagram%22%2C%22is%22%29%29%2C%0D%0Alocate%289%2C17.2%2Cmatrix%281%2C2%2C%22to%22%2C%22scale%22%29%29%0D%0A%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Note that I'm copying the arrow style (to the best of my ability) shown in the diagram link. Your teacher/textbook should use different arrow styles for a different set of parallel segments. For instance AB & ED could have single arrow, while AF & DC have double arrows.\r<BR>\n" );
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document.write( "The arrows on the segments tell us the following <BR>\n" );
document.write( "AB is parallel to DE<BR>\n" );
document.write( "BC is parallel to AD<BR>\n" );
document.write( "AF is parallel to DC\r<BR>\n" );
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document.write( "Consequently it means ADEB and ADCF are parallelograms.<BR>\n" );
document.write( "The opposite sides of a parallelogram are congruent, which leads to AD = BE = 4 and AD = FC = 4.<BR>\n" );
document.write( "BC = BE+EF+FC = 4+3+4 = 11\r<BR>\n" );
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document.write( "h = height of trapezoid ABCD<BR>\n" );
document.write( "area of trapezoid = 0.5*height*(base1+base2)<BR>\n" );
document.write( "area of trapezoid ABCD = 0.5*h*(AD+BC)<BR>\n" );
document.write( "135 = 0.5*h*(AD+BC)<BR>\n" );
document.write( "0.5*h*(4+11) = 135<BR>\n" );
document.write( "7.5h = 135<BR>\n" );
document.write( "h = 135/7.5<BR>\n" );
document.write( "h = 18<BR>\n" );
document.write( "Trapezoid ABCD has height 18 cm.<BR>\n" );
document.write( "This is also the height of triangle ABF when working with base BF.\r<BR>\n" );
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document.write( "Focus on triangle ABF. <BR>\n" );
document.write( "To avoid clutter, you can create a quick sketch of triangle ABF off to the side somewhere. <BR>\n" );
document.write( "area = 0.5*base*height<BR>\n" );
document.write( "area = 0.5*BF*h<BR>\n" );
document.write( "area = 0.5*(BE+EF)*h<BR>\n" );
document.write( "area = 0.5*(4+3)*18<BR>\n" );
document.write( "area = 63 square cm\r<BR>\n" );
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document.write( "Notice that triangles ABF and NEF are similar triangles (use the Angle Angle Theorem; I will leave this as an exercise for the student).\r<BR>\n" );
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document.write( "Let's compare the horizontal portions of each triangle.<BR>\n" );
document.write( "The ratio of segment EF to BF is 3:7<BR>\n" );
document.write( "Square both parts to get 9:49 which is the ratio of their areas.<BR>\n" );
document.write( "This new ratio means that if triangle NEF had area 9 square cm, then triangle ABF would have an area of 49 square cm.\r<BR>\n" );
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document.write( "In other words, triangle NEF takes up 9/49 of triangle ABF's area.<BR>\n" );
document.write( "The remaining portion is 1 - (9/49) = 40/49 which is what trapezoid ANEB takes up.\r<BR>\n" );
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document.write( "area of ANEB = (40/49)*(area of triangle ABF) <BR>\n" );
document.write( "= (40/49)*(63)<BR>\n" );
document.write( "= (360/7) square cm<BR>\n" );
document.write( "= 51.42857 square cm approximately\r<BR>\n" );
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document.write( "Now focus on triangle ABC.<BR>\n" );
document.write( "area = 0.5*base*height<BR>\n" );
document.write( "area = 0.5*BC*h<BR>\n" );
document.write( "area = 0.5*11*18<BR>\n" );
document.write( "area = 99 square cm\r<BR>\n" );
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document.write( "Triangles MEC and ABC are similar.<BR>\n" );
document.write( "The ratio of the horizontal pieces EC to BC is 7:11<BR>\n" );
document.write( "Both parts square to 49:121 which is the ratio of the areas MEC to ABC.<BR>\n" );
document.write( "Triangle MEC takes up 49/121 of triangle ABC.<BR>\n" );
document.write( "Trapezoid AMEB takes up 1 - (49/121) = 72/121 of triangle ABC's area.\r<BR>\n" );
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document.write( "area of AMEB = (72/121)*(area of triangle ABC)<BR>\n" );
document.write( "= (72/121)*(99)<BR>\n" );
document.write( "= (648/11) square cm<BR>\n" );
document.write( "= 58.90909 square cm approximately\r<BR>\n" );
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document.write( "There are a lot of things to keep track of. <BR>\n" );
document.write( "But the two key important things are: <BR>\n" );
document.write( "area of trapezoid AMEB = (648/11) square cm<BR>\n" );
document.write( "area of trapezoid ANEB = (360/7) square cm\r<BR>\n" );
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document.write( "Subtracting those areas will get us the area of triangle AMN.<BR>\n" );
document.write( "area AMN = (area AMEB) - (area ANEB)<BR>\n" );
document.write( "area AMN = (648/11) - (360/7)<BR>\n" );
document.write( "area AMN = (648/11)*(7/7) - (360/7)*(11/11)<BR>\n" );
document.write( "area AMN = (648*7)/(11*7) - (360*11)/(7*11)<BR>\n" );
document.write( "area AMN = 4536/77 - 3960/77<BR>\n" );
document.write( "area AMN = (4536 - 3960)/77<BR>\n" );
document.write( "area AMN = <font color=red>(576/77) square cm exactly</font><BR>\n" );
document.write( "area AMN = <font color=red>7.480519480519 square cm approximately</font><BR>\n" );
document.write( "Round this approximate value however needed.\r<BR>\n" );
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document.write( "I have confirmed this answer is correct using <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> \r<BR>\n" );
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document.write( "A similar problem is found <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/word/geometry/Geometry_Word_Problems.faq.question.1187703.html\">here</a><BR>\n" );
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