document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1149604>Question 1149604</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the diagram below, AC is the median of triangle ABD. Find the area of triangle ABD\r<BR>\n" );
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document.write( "Diagram: https://imgur.com/a/vr9cZmy </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #770937 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=770937\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "Given image<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/BBREqMU.png\"><BR>\n" );
document.write( "Let point E be the midpoint of segment BC. The red triangle AEC has unknown height h and unknown base x<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/UFIdzie.png\"><BR>\n" );
document.write( "Focusing solely on triangle AEC, we can use the pythagorean theorem to say\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+%2B+b%5E2+=+c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2+%2B+b%5E2+=+c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+%2B+x%5E2+=+17%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+%2B+x%5E2+=+17%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+%2B+x%5E2+=+289 ALIGN=MIDDLE  ALT=\"h%5E2+%2B+x%5E2+=+289\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+289-x%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+=+289-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > subtract x^2 from both sides\r<BR>\n" );
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document.write( "We'll use the last equation later on\r<BR>\n" );
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document.write( "Mark triangle AED with another color. I'll use green. This triangle also has height h, but the base is now 3x<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/tUFe5yf.png\"><BR>\n" );
document.write( "The 3x is from the fact that\r<BR>\n" );
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document.write( "EC = x<BR>\n" );
document.write( "CD = BC = 2x<BR>\n" );
document.write( "ED = EC+CD = x+2x = 3x\r<BR>\n" );
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document.write( "Once again, use the pythagorean theorem, but focus solely on triangle AED.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+%2B+b%5E2+=+c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2+%2B+b%5E2+=+c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+%2B+%283x%29%5E2+=+27%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+%2B+%283x%29%5E2+=+27%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+%2B+9x%5E2+=+729 ALIGN=MIDDLE  ALT=\"h%5E2+%2B+9x%5E2+=+729\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=289-x%5E2+%2B+9x%5E2+=+729 ALIGN=MIDDLE  ALT=\"289-x%5E2+%2B+9x%5E2+=+729\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2 ALIGN=MIDDLE  ALT=\"h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=289-x%5E2 ALIGN=MIDDLE  ALT=\"289-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (we're using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+289-x%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+=+289-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > found earlier)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=289%2B8x%5E2+=+729 ALIGN=MIDDLE  ALT=\"289%2B8x%5E2+=+729\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E2+=+729-289 ALIGN=MIDDLE  ALT=\"8x%5E2+=+729-289\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 289 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E2+=+440 ALIGN=MIDDLE  ALT=\"8x%5E2+=+440\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+440%2F8 ALIGN=MIDDLE  ALT=\"x%5E2+=+440%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 8\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+55 ALIGN=MIDDLE  ALT=\"x%5E2+=+55\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%2855%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%2855%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Apply square root to both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+=+4sqrt%2855%29 ALIGN=MIDDLE  ALT=\"4x+=+4sqrt%2855%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by 4\r<BR>\n" );
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document.write( "The base of triangle ABD is exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Asqrt%2855%29 ALIGN=MIDDLE  ALT=\"4%2Asqrt%2855%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > units long\r<BR>\n" );
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document.write( "Use that info to find the following value for h\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+289-x%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+=+289-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+289-55 ALIGN=MIDDLE  ALT=\"h%5E2+=+289-55\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+234 ALIGN=MIDDLE  ALT=\"h%5E2+=+234\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "----------------------\r<BR>\n" );
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document.write( "Now compute the area of triangle ABD\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area+=+%281%2F2%29%2A%28base%29%2A%28height%29 ALIGN=MIDDLE  ALT=\"Area+=+%281%2F2%29%2A%28base%29%2A%28height%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area+=+%281%2F2%29%2A4sqrt%2855%29%2Asqrt%28234%29 ALIGN=MIDDLE  ALT=\"Area+=+%281%2F2%29%2A4sqrt%2855%29%2Asqrt%28234%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area+=+2%2Asqrt%2855%29%2Asqrt%28234%29 ALIGN=MIDDLE  ALT=\"Area+=+2%2Asqrt%2855%29%2Asqrt%28234%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area+=+2%2Asqrt%2855%2A234%29 ALIGN=MIDDLE  ALT=\"Area+=+2%2Asqrt%2855%2A234%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area+=+2%2Asqrt%2812870%29 ALIGN=MIDDLE  ALT=\"Area+=+2%2Asqrt%2812870%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area+=+2%2Asqrt%289%2A1430%29 ALIGN=MIDDLE  ALT=\"Area+=+2%2Asqrt%289%2A1430%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area+=+2%2Asqrt%289%29%2Asqrt%281430%29 ALIGN=MIDDLE  ALT=\"Area+=+2%2Asqrt%289%29%2Asqrt%281430%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area+=+2%2A3%2Asqrt%281430%29 ALIGN=MIDDLE  ALT=\"Area+=+2%2A3%2Asqrt%281430%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area+=+6%2Asqrt%281430%29 ALIGN=MIDDLE  ALT=\"Area+=+6%2Asqrt%281430%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <font color=red size=4>Exact area</font>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area+=+226.892 ALIGN=MIDDLE  ALT=\"Area+=+226.892\"   DISABLED_style= \"max-width:90%; height:auto;\" > <font color=red size=4>Approximate area</font>\r<BR>\n" );
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document.write( "Units are in square cm, often written as <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" );
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