document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=233654>Question 233654</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what  is the ratio of the area of square  to the area of an equilateral triangle<BR>\n" );
document.write( "if they have the same perimeter ?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #172477 by </B><A HREF=/tutors/aboutme.mpl?userid=ptaylor><B>ptaylor(2198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ptaylor><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=ptaylor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=172477\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let s=length of a side of the equilateral triangle<BR>\n" );
document.write( "Area (A) of a triangle equals (1/2)base(b)*height(h)=(1/2)bh<BR>\n" );
document.write( "Height of an equilateral triangle with sides s units long is (using the Pythagorean theorm):<BR>\n" );
document.write( "s^2=((1/2)s)^2+h^2 or h^2=s^2-(1/4)s^2=(3/4)s^2 and<BR>\n" );
document.write( "h=(sqrt(3))/2)s  so area would be<BR>\n" );
document.write( "A=(1/2)s*(sqrt(3)/2)s=s^2(sqrt(3))/4)--------------eq1 (area of equilateral triangle with sides s units long).<BR>\n" );
document.write( "Let l=length of the sides of the square<BR>\n" );
document.write( "Area (A) of the square=l^2------------------------eq2\r<BR>\n" );
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document.write( "Now we are told that the perimeter of the triangle and square are equal.  That means that 3s=4l or l=(3/4)s. Now we'll substitute this into eq2 and we get<BR>\n" );
document.write( "Area of square=((3/4)s)^2=(9/16)s^2\r<BR>\n" );
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document.write( "Now for the ratio:<BR>\n" );
document.write( "So (Area of square)/(Area of triangle)=(9/16)s^2/((sqrt(3))/4)s2)<BR>\n" );
document.write( "The s^2 cancel on the right side so we end up with:\r<BR>\n" );
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document.write( "(Area of square)/(Area of triangle)=(9/16)/sqrt(3)/4) multiply numerator and denomionator by 4/(sqrt(3)) (Note: this will get rid of the complex fraction)\r<BR>\n" );
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document.write( "(Area of square)/(Area of triangle)=((9/16)*(4/(sqrt(3))/1 and this reduces to:<BR>\n" );
document.write( "9/(4(sqrt(3)) now we multiply the numerator and denominator by sqrt(3) and we get:9(sqrt(3))/4*3=(9/12)sqrt(3) or (3/4)sqrt(3)\r<BR>\n" );
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document.write( "So,(Area of square)/(Area of triangle)=(3/4)sqrt(3)---shud be the ans\r<BR>\n" );
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document.write( "Hope this helps---ptaylor<BR>\n" );
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