document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1060919>Question 1060919</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In triangle ABC, E and F are midpoints of sides line AB and line AC, respectively; and H and I trisect the side BC. If the area of triangle ABC is 120, what is the area of triangle EFJ?\r<BR>\n" );
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document.write( "Image is in the link: http://prntscr.com/dhjxtm </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #675852 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=675852\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C400%2C-0.5%2C9.5%2C-1%2C7%2C%0D%0Atriangle%280%2C0%2C9%2C0%2C4%2C6%29%2Cred%28line%282%2C3%2C6.5%2C3%29%29%2C%0D%0Agreen%28line%282%2C3%2C6%2C0%29%29%2Cgreen%28line%283%2C0%2C6.5%2C3%29%29%2C%0D%0Alocate%283.95%2C6.3%2CA%29%2Clocate%28-0.1%2C0%2CB%29%2Clocate%289%2C0%2CC%29%2C%0D%0Alocate%281.8%2C3.2%2CE%29%2Clocate%286.6%2C3.2%2CF%29%2C%0D%0Alocate%282.9%2C0%2CH%29%2C+locate%285.9%2C0%2CI%29%2Clocate%284%2C1.4%2CJ%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area of BCA is (1/2)(AB)*h,<BR>\n" );
document.write( "with h being the height relative to base BC,<BR>\n" );
document.write( "or (in other words, the distance from BC to A).<BR>\n" );
document.write( "Because E and F are midpoints of sides AB and AC,<BR>\n" );
document.write( "EFA and BCA are similar, EFA being a (1/2) scale version of BCA,<BR>\n" );
document.write( "the height of EFA (distance from EF to A) being (1/2)h.<BR>\n" );
document.write( "That makes the distance from AB to EF also (1/2)h,<BR>\n" );
document.write( "and makes the area of EFA 1/4 of the area of BCA.<BR>\n" );
document.write( "(That is (1/4)120=30 for the area of EFA).<BR>\n" );
document.write( "Angles EJF and HJI are congruent because they are vertical angles,<BR>\n" );
document.write( "formed by the intersection of lines EI and HF.<BR>\n" );
document.write( "Angles JIH=EIH=EIB and FEJ=FEI are also congruent because they are alternate interior angles.<BR>\n" );
document.write( "formed between parallel lines BC and EF by trasversal EI.<BR>\n" );
document.write( "That makes triangles HIJ and EFJ similar triangles.<BR>\n" );
document.write( "The length of their bases, EF={1/2)AB, and HI=(1/3)AB are in a 3:2 ratio.<BR>\n" );
document.write( "The ratio of the heights relative to those bases is also 3/2.<BR>\n" );
document.write( "Their heights add up to (1/2)h, the distance between AB and EF.<BR>\n" );
document.write( "That makes the height of HIJ (2/5)(1/2)h=(2/10)h,<BR>\n" );
document.write( "and the height of EFJ (3/5)(1/2)h=(3/10)h.<BR>\n" );
document.write( "The area of EFJ, with a base (1/2) of the base of BCA,<BR>\n" );
document.write( "and a height (3/10) of the height of BCA,<BR>\n" );
document.write( "is (1/2)(3/10)=3/20 of the area of BCA.<BR>\n" );
document.write( "So Area of EFJ=(3/20)120=3*6=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2818%29 ALIGN=MIDDLE  ALT=\"highlight%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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