document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1027679>Question 1027679</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An equilateral triangle circumscribes a circle with radius 4. Find the height of the triangle.\r<BR>\n" );
document.write( "\n" );
document.write( "How was the answer 12 units? Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #642912 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=642912\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> each angle of an equilateral triangle is equal to 60 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if you drop a perpendicular from each of the vertices to the opposite side, you will find that intersection of these perpendiculars will be the center of the inscribed circle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you will also find that this separates the triangle into 6 separate smaller triangles.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you will also find that each vertex of the equilateral triangle will have had their angles cut exactly in half, with each of the halves equal to 30 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the six triangles formed are all right triangles because the perpendiculars form right angles with the opposite side of the vertex from which they were formed.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "all of the triangle are therefore 30,60,90 right triangles.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "one leg of each of the six triangles formed is equal to the radius of the inscribed circle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this makes one leg of each right triangle equal to 4 units.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since the opposite angle is 30 degrees, and since the sine of 30 degrees is 1/2, we can use that fact to find the length of the hypotenuse of each of those triangles.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "we'll use one of those triangle to show you what happens.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in the diagram, the altitude of the equilateral triangle we chose is the line segment BD.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the triangle formed that we chose is triangle BEF.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the point E is the center of the circle and also the intersection of the 3 perpendicular we dropped from the vertices of the equilateral triangle, which is triangle ABC.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the 30 degree angles we chose is angle EBF.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the opposite side of that angle is EF which is equal to 4 units.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the hypotenuse of that triangle is EB.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "we are working with triangle BEF.<BR>\n" );
document.write( "the 30 degree angle of that triangle is angle EBF.<BR>\n" );
document.write( "the opposite side to that 30 degree angle is side EF.<BR>\n" );
document.write( "the hypotenuse of that triangle is BE.<BR>\n" );
document.write( "the 90 degree angle of that triangle is angle BFE.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "we know that the sine of an angle is equal to opposite / hypotenuse.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "therefore sin(EBF) = EF / BE.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since angle EBF = 30 degrees and since EF = 4, we get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sin(30) = 4 / BE.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for BE to get BE = 4 / sin(30) = 8.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "we know that BE is part of BD.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "we know that the other part of BD is ED.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "we know that BE is equal to 8 and ED is equal to 4, therefore the length of BD is equal to 12.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the line segment BD is the height of the equilateral triangle ABC, therefore we have found the height of triangle ABC.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "it is 12 units.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the diagram is shown below:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/040204.jpg\" alt=\"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );