document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=328006>Question 328006</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if the measures of angles of an triangle  in the ratio of 1:2:3, and if the perimeter of the triangle is 30 + 10 sqrt(3), what is the length of the smallest side ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #234921 by </B><A HREF=/tutors/aboutme.mpl?userid=rapaljer><B>rapaljer(4671)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rapaljer><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=rapaljer><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=234921\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If the angles are in a ratio of 1:2:3, then let x = smallest angle.\r<BR>\n" );
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document.write( "The other angles are 2x and 3x respectively, so <BR>\n" );
document.write( "x + 2x + 3x=180<BR>\n" );
document.write( "6x=180<BR>\n" );
document.write( "x=30<BR>\n" );
document.write( "2x=60<BR>\n" );
document.write( "3x=90\r<BR>\n" );
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document.write( "This is a 30-60-90 triangle, so the sides are in a ratio of 1:2:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Let y= first side<BR>\n" );
document.write( "2y= second side (hypotenuse of right triangle!)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"y%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= third side\r<BR>\n" );
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document.write( "Perimeter=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y%2B+2y+%2B+y%2Asqrt%283%29=30%2B10%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"+y%2B+2y+%2B+y%2Asqrt%283%29=30%2B10%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3y%2B+y%2Asqrt%283%29=+30%2B+10%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"3y%2B+y%2Asqrt%283%29=+30%2B+10%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore, y=10.\r<BR>\n" );
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document.write( "The smallest side is 10, the second side is 20, and the third side is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"10%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The perimeter checks!!\r<BR>\n" );
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document.write( "For additional help on GEOMETRY, please see my own website.  Click on my tutorname \"rapaljer\" anywhere in algebra.com, and look for \"Topics from Geometry\" near the bottom of the page.  \r<BR>\n" );
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document.write( "Dr. Robert J. Rapalje, Retired<BR>\n" );
document.write( "Seminole State College of Florida<BR>\n" );
document.write( "Altamonte Springs Campus\r<BR>\n" );
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