document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207466>Question 1207466</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The regular hexagon ABCDEF has sides of length 2. The point P is the midpoint of AB. Q is the midpoint of BC and so on. Find the area of the hexagon PQRSTU. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845349 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=845349\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the area of a hexagon is equal to 3 * sqrt(3) / 2 * a^2, where a is the length of one side of the regular hexagon.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the area of your original hexagon is equal to 3 * sqrt(3) / 2 * 2^2 = 10.39230485.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when you connect the midpoints of two adjacent triangles formed by your hexagon, then the side of that new hexagon is equal to 1.732050808.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the area of the new hexagon is equal to 3 * sqrt(3) / 2 * 1.732050808^2 = 7.794228634.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that should be your answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the only question you might have is how i derived the value of the side of the new hexagon formed by the midpoints of the original hexagon.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that takes a little explaining.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "see my worksheet below.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2024/060308.jpg\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in the diagram:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "K is the center of the hexagon.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "AKB is one triangle formed from the center of the hexagon to the sides of the hexagon.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "BKC is an adjacent triangle formed from the center of the hexagon to the sides of the hexagon.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "KP is a line from the center of the hexagon to the midpoint of one side of the hexagon.<BR>\n" );
document.write( "KQ is a line from the center of the hexagon to the midpoint of an adjacent side of the hexagon.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the length of AB is 2 and the length of BC is 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you form one side of the new hexagon by connecting point P to Q.<BR>\n" );
document.write( "that forms triangle PKQ which is one of the 6 triangles of the new hexagon.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "angle AKB is split in two by the line KP which terminates at the midpoint of line AB.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "AK is the hypotenuse of right triangle AKP.<BR>\n" );
document.write( "angle AKP is 30 degrees.<BR>\n" );
document.write( "sine(30) = 1 / AK<BR>\n" );
document.write( "AK = 1 / sine(30) = 2.<BR>\n" );
document.write( "that's how the length of AK was derived to be 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "PK = 1.732050808.<BR>\n" );
document.write( "that's found because tan(30) = 1 / PK<BR>\n" );
document.write( "PK = 1 / tan(30) which is equal to 1.732050808.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when you connect point P to Q, that line intersects with line KB at point R.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that forms right triangle PRB.<BR>\n" );
document.write( "the hypotenuse of that right triangle is PB whose length is equal to 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "cosine (RPB) = PR / 1<BR>\n" );
document.write( "solve for PR to get PR = 1 * cosine(RPB)<BR>\n" );
document.write( "that becomes cosine(30) = .8660254038.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the length of PQ is twice that to be equal to 1.732050808.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's one side of your new hexagon.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "area of your new hexagon is equal to 3 * sqrt(3) / 2 * 1.732050808 = 7.794228634.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that should be your answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );