document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=201375>Question 201375</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The middlepoints of the side of a triangle are joined forming a second triangle.Again a third triangle is formed by joining the middles points of this second triangle and this process is repeated infinitely.If the perimeter and area of the outer triangle are P and A respectively,What will the sum of perimeters of the triangles thus formed? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #151613 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=151613\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "If you join the mid-points of two sides of a triangle then you form a smaller similar triangle the sides of which are each one-half the measure of the sides of the original triangle, hence the perimeter of the smaller triangle is also one-half the perimeter of the original.  If the perimeter of the first triangle is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P\">, then the perimeter of the smaller triangle is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{P}{2}\">.  The next one would be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{P}{4}\">, and so on.\r<BR>\n" );
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document.write( "Therefore the sum of the perimeters would be:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P + P\sum_{k=1}^\infty\,\frac{1}{2^k}\">\r<BR>\n" );
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document.write( "But since\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{n\rightarrow\infty}\ \sum_{k=1}^n\,\frac{1}{2^k} = 1\">\r<BR>\n" );
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document.write( "The sum of the perimeters becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P + P\cdot 1 = 2P\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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