document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=203384>Question 203384</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the area of an equilateral triangle whose apothem is 8 yards? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #153456 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=153456\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> to get the apothemn, you inscribe a circle inside the equilateral triangle.<BR>\n" );
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document.write( "the radius of that circle will intersect each side exactly in the middle and will be perpendicular to each side at that point.<BR>\n" );
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document.write( "if you draw a line from the center of that circle to each of the corners of the triange, then the center of that circle divides the equilateral triangle into 3 separate triangles.  each one of those separate triangles has a side of the equilateral triangle as it's base and the lines drawn from the center of the circle to each corner of the triangle as it's other two sides.  <BR>\n" );
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document.write( "each one of those triangles has a 120 degree angle at the center of the circle and two 30 degree angles at each end.<BR>\n" );
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document.write( "the apothem in each in each of those 3 separate triangles is the perpendicular bisector each of them.<BR>\n" );
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document.write( "what you have is 6 separate right triangles formed from those 3 separate triangles that have one angle of 60 degrees, one angle of 90 degrees, and one angle of 30 degrees each.<BR>\n" );
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document.write( "since the apothem is the height of each of those 6 right triangles, then the base of each of those 6 right triangles is equal to the apothem of each of those right triangles * cot(30) where 30 represents 30 degrees.<BR>\n" );
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document.write( "this is derivede as follows:<BR>\n" );
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document.write( "cot 30) = adjacent side / opposite side = base / apothem.<BR>\n" );
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document.write( "cot(30) = 1/tan(30) = 1/.577350269 = 1.732050808<BR>\n" );
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document.write( "cot(30) = base / apothem = base/8 where 8 is the length of the apothem.<BR>\n" );
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document.write( "multiply both sides of this equation by 8 to get:<BR>\n" );
document.write( "base of each right triangle = cot(30) * 8 = 1.732050808 * 8 = 13.85640646<BR>\n" );
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document.write( "since base of each right triangle = 1/2 the length of each side of the equilateral triangle, then each side of the equilateral triangle equals 2 * 13.85640646 = 27.71281292<BR>\n" );
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document.write( "the area of the equilateral triangle should be able to be derived in a couple of ways.<BR>\n" );
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document.write( "one way is to get the area of each of the right triangles and multiply by 6.<BR>\n" );
document.write( "area of each right triangle is 1/2 * 8 * 13.85640646  = 55.42562584 * 6 = 332.5537551<BR>\n" );
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document.write( "another way is to use the formula for the area of an equilateral triangle which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F4+%2A+s%5E2 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F4+%2A+s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%2F4+%2A+%2827.71281292%29%5E2+=+332.5537551 ALIGN=MIDDLE  ALT=\"sqrt%283%29%2F4+%2A+%2827.71281292%29%5E2+=+332.5537551\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "we get the same answer both ways which is a good sign.<BR>\n" );
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document.write( "the apothem is 8 (given).<BR>\n" );
document.write( "each side of the equiateral triangle is 27.71281292 yards long.<BR>\n" );
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document.write( "your answer is:<BR>\n" );
document.write( "area of the equilateral triangle is 332.5537551 square yards.<BR>\n" );
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document.write( "to draw your own picture, do the following.<BR>\n" );
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document.write( "equilateral triangle = ABC<BR>\n" );
document.write( "D = center of inscribed circle.<BR>\n" );
document.write( "DE is perpendicular bisector of BC<BR>\n" );
document.write( "DF is perpendicular bisector of AC<BR>\n" );
document.write( "DG is perpendicular bisector of AB<BR>\n" );
document.write( "Draw DB, DC, DA<BR>\n" );
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document.write( "your 3 separate triangles are:<BR>\n" );
document.write( "BDC, BDA, ADC<BR>\n" );
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document.write( "your 6 separate right triangles are:<BR>\n" );
document.write( "BDE, CDE (from BDC)<BR>\n" );
document.write( "BDG, ADG (from BDA)<BR>\n" );
document.write( "CDF, ADF (from ADC)<BR>\n" );
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document.write( "your apothems are:<BR>\n" );
document.write( "DE, DF, DG<BR>\n" );
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document.write( "if you need a picture, let me know and i'll work one up and send it to you.  <BR>\n" );
document.write( "try drawing your own first.<BR>\n" );
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