document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=981361>Question 981361</A>: <I><SPAN STYLE=\"word-break: break-all;\"> D and E ARE POINTS ON THE SIDES AB AND AC RESPECTIVELY OF TRIANGLE ABC SUCH THAT DE IS PARALLEL TO BC, AND AD:DB = 4:5. CD AND BE INTERSECT EACH OTHER AT F. FIND THE RATIO OF THE AREAS OF TRIANGLE DEF AND TRIANGLE BCF . </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #602494 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=602494\">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%28300%2C300%2C-1%2C10%2C-1%2C10%2C%0D%0Atriangle%280%2C0%2C9%2C0%2C6%2C9%29%2C%0D%0Atriangle%283.33%2C5%2C7.33%2C5%2C9%2C0%29%2C%0D%0Atriangle%283.33%2C5%2C7.33%2C5%2C0%2C0%29%2C%0D%0Alocate%28-0.25%2C0%2CB%29%2Clocate%289%2C0%2CC%29%2Clocate%285.8%2C9.5%2CA%29%2C%0D%0Aarrow%283.33%2C5%2C5.33%2C5%29%2Carrow%282.5%2C0%2C4.5%2C0%29%2C%0D%0Alocate%282.9%2C5.25%2CD%29%2Clocate%287.4%2C5.25%2CE%29%2Clocate%284.95%2C3.4%2CF%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "We can prove that<BR>\n" );
document.write( "triangles ADE and ABC are similar and their corresponding sides' lengths are in the ratio 4:9,<BR>\n" );
document.write( "and that<BR>\n" );
document.write( "triangles DEF and CBF are similar and their corresponding sides' lengths are in the ratio 4:9.<BR>\n" );
document.write( "Since for similar figures the ratio of the areas is the square of the ratio of their corresponding sides' lengths,<BR>\n" );
document.write( "the ratio of areas of DEF and CBF is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2F9%29%5E2=highlight%2816%2F81%29 ALIGN=MIDDLE  ALT=\"%284%2F9%29%5E2=highlight%2816%2F81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The ratio of areas of triangle DEF and triangle BCF is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2816%3A81%29 ALIGN=MIDDLE  ALT=\"highlight%2816%3A81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "PROOF DETAILS:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD%2FDB=4%2F5=0.8 ALIGN=MIDDLE  ALT=\"AD%2FDB=4%2F5=0.8\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD=0.8DB ALIGN=MIDDLE  ALT=\"AD=0.8DB\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD%2FAB=AD%2F%28AD%2BDB%29=0.8DB%2F%280.8DB%2BDB%29=0.8DB%2F%221.8+DB%22=0.8%2F1.8=8%2F18=4%2F9 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Angles BAC and DAE are congruent because they are the same angle.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Angles ABC and ADE are congruent because they are corresponding angles formed by parallel lines BC and DE with transversal AB.<BR>\n" );
document.write( "Angles ACB and AED are congruent because they are corresponding angles formed by parallel lines BC and DE with transversal AC.<BR>\n" );
document.write( "Triangles ABC and ADE are similar, because they have three pairs of congruent angles.<BR>\n" );
document.write( "Since the triangles are similar the ratios of corresponding sides are the same,<BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD%2FAB=4%2F9 ALIGN=MIDDLE  ALT=\"AD%2FAB=4%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > , then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DE%2FBC=4%2F9 ALIGN=MIDDLE  ALT=\"DE%2FBC=4%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "Angle DEF is congruent with angle CBF because they are alternate interior angles formed by parallel lines BC and DE with transversal EB.<BR>\n" );
document.write( "Angle EDF is congruent with angle BCF because they are alternate interior angles formed by parallel lines BC and DE with transversal EB.<BR>\n" );
document.write( "Angle DFE is congruent with angle CF because they are vertical angles.<BR>\n" );
document.write( "Triangles DEF and CBF are similar, because they have three pairs of congruent angles.<BR>\n" );
document.write( "Since the triangles are similar the ratios of corresponding sides are the same:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DE%2FBC=4%2F9 ALIGN=MIDDLE  ALT=\"DE%2FBC=4%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since for similar figures the ratio of the areas is the square of the ratio of their corresponding sides' lengths,<BR>\n" );
document.write( "the ratio of areas of DEF and CBF is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2F9%29%5E2=highlight%2816%2F81%29 ALIGN=MIDDLE  ALT=\"%284%2F9%29%5E2=highlight%2816%2F81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The ratio of areas of triangle DEF and triangle BCF is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2816%3A81%29 ALIGN=MIDDLE  ALT=\"highlight%2816%3A81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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