document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=119312>Question 119312</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the area of an equilateral triangle whose altitude is 9? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #87407 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=87407\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If you construct the altitude of an equilateral triangle, you create a 30-60-90 degree right triangle.\r<BR>\n" );
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document.write( "If you know that the proportions of the sides of a 30-60-90 right triangle are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, fine.  Otherwise, read the following discussion.\r<BR>\n" );
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document.write( "Assume you have an equilateral triangle with sides that measure 1.  Construct the altitude.  The altitude bisects one of the sides, so the short leg of the resulting 30-60-90 triangle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   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=sqrt%281%5E2-%281%2F2%29%5E2%29=sqrt%283%2F4%29=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%281%5E2-%281%2F2%29%5E2%29=sqrt%283%2F4%29=sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  So the sides are in proportion <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "In the given equilateral triangle, the hypotenuse of the 30-60-90 right triangle is equal to the base of the equilateral triangle because the equilateral triangle's sides are all equal.  So to find the measure of the base of the equilateral triangle we need the proportion:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2F%28sqrt%283%29%2F2%29=x%2F1 ALIGN=MIDDLE  ALT=\"9%2F%28sqrt%283%29%2F2%29=x%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Cross-multiply:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%283%29%2F2%29x=9 ALIGN=MIDDLE  ALT=\"%28sqrt%283%29%2F2%29x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"2%2Fsqrt%283%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=x=18%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"x=18%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Rationalize the denominator:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2818%2Fsqrt%283%29%29%28sqrt%283%29%2Fsqrt%283%29%29=%2818%2Asqrt%283%29%29%2F3=6%2Asqrt%283%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The area of the triangle is given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28bh%29%2F2 ALIGN=MIDDLE  ALT=\"A=%28bh%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where b is the base and h is the height or altitude.  Substituting our given and calculated values:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28%286%2Asqrt%283%29%29%2A%289%29%29%2F2=27%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"A=%28%286%2Asqrt%283%29%29%2A%289%29%29%2F2=27%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > square units.\r<BR>\n" );
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document.write( "Hope that helps,<BR>\n" );
document.write( "John </SPAN>\n" );
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