document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117949>Question 1117949</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two sides and an angle&#8203; (SSA) of a triangle are given. Determine whether the given measurements produce one&#8203; triangle, two&#8203; triangles, or no triangle at all. Solve each triangle that results. <BR>\n" );
document.write( "a=13<BR>\n" );
document.write( "c=14<BR>\n" );
document.write( "A=55 degrees.\r<BR>\n" );
document.write( "\n" );
document.write( "We're finding all remaining sides and angles.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #733132 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=733132\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We can draw side <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB ALIGN=MIDDLE  ALT=\"AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=14 ALIGN=MIDDLE  ALT=\"c=14\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "and draw a ray (arrow) that will contain side <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC ALIGN=MIDDLE  ALT=\"AC\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "starting from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > and making an angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=55%5Eo ALIGN=MIDDLE  ALT=\"A=55%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > with side <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB ALIGN=MIDDLE  ALT=\"AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > is at a distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=13 ALIGN=MIDDLE  ALT=\"a=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "so it is going to be where the circle with center <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > and radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" > crosses the ray (arrow) .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C15%2C-1%2C15%2C%0D%0Aline%280%2C0%2C14%2C0%29%2Carrow%280%2C0%2C10%2C14.28%29%2C%0D%0Agreen%28arc%2814%2C0%2C26%2C26%2C180%2C280%29%29%2C%0D%0Agreen%28arrow%2814%2C6.5%2C14%2C0%29%29%2C%0D%0Agreen%28arrow%2814%2C6.5%2C14%2C13%29%29%2C%0D%0Ared%28arc%280%2C0%2C6%2C6%2C-55%2C0%29%29%2C%0D%0Alocate%282.6%2C2.6%2Cred%28A=55%5Eo%29%29%2C%0D%0Alocate%286.5%2C0%2Cc=14%29%2C%0D%0Alocate%2814%2C7%2Cgreen%2813%29%29%2C%0D%0Alocate%28-0.2%2C0%2CA%29%2Clocate%2813.8%2C0%2CB%29%2C%0D%0Acircle%281.09%2C1.56%2C0.2%29%2Ccircle%288.12%2C11.59%2C0.2%29%2C%0D%0Alocate%280.1%2C2%2C%22C+%3F%22%29%2Clocate%288%2C11.35%2C%22C+%3F%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The problem is that in this case, we find two possible locations for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so  the given measurements produce two&#8203; triangles.<BR>\n" );
document.write( "From the drawing above, we can see that if the measurement for side <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC ALIGN=MIDDLE  ALT=\"BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=15 ALIGN=MIDDLE  ALT=\"a=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we would have one&#8203; triangle,<BR>\n" );
document.write( "and if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC ALIGN=MIDDLE  ALT=\"BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > was very short (as in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=5 ALIGN=MIDDLE  ALT=\"a=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > for example),<BR>\n" );
document.write( "we would have no triangle at all.<BR>\n" );
document.write( " <BR>\n" );
document.write( "As we have measurements for one angle and the opposite side, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28A=55%5Eo%2Ca=13%29 ALIGN=MIDDLE  ALT=\"system%28A=55%5Eo%2Ca=13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we can solve for the two&#8203; triangles using Law of Sines:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A%29%2Fa=sin%28B%29%2Fb=sin%28C%29%2Fc ALIGN=MIDDLE  ALT=\"sin%28A%29%2Fa=sin%28B%29%2Fb=sin%28C%29%2Fc\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fsin%28A%29=b%2Fsin%28B%29=c%2Fsin%28C%29 ALIGN=MIDDLE  ALT=\"a%2Fsin%28A%29=b%2Fsin%28B%29=c%2Fsin%28C%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Having <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we start by using<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A%29%2Fa=sin%28C%29%2Fc ALIGN=MIDDLE  ALT=\"sin%28A%29%2Fa=sin%28C%29%2Fc\"   DISABLED_style= \"max-width:90%; height:auto;\" > to try to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.819152%2F13=sin%28C%29%2F14 ALIGN=MIDDLE  ALT=\"0.819152%2F13=sin%28C%29%2F14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14%2A0.819152%2F13=sin%28C%29 ALIGN=MIDDLE  ALT=\"14%2A0.819152%2F13=sin%28C%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28C%29=0.882164 ALIGN=MIDDLE  ALT=\"sin%28C%29=0.882164\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "My calculator says that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=61.9%5Eo ALIGN=MIDDLE  ALT=\"C=61.9%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a solution.<BR>\n" );
document.write( "That gives us<BR>\n" );
document.write( "TRIANGLE #1, with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=180%5Eo-%28A%2BC%29=180%5Eo-%2855%5Eo%2B61.9%5Eo%29=180%5Eo-116.9%5Eo=63.1%5Eo ALIGN=MIDDLE  ALT=\"B=180%5Eo-%28A%2BC%29=180%5Eo-%2855%5Eo%2B61.9%5Eo%29=180%5Eo-116.9%5Eo=63.1%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Now that we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we can use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fsin%28A%29=b%2Fsin%28B%29 ALIGN=MIDDLE  ALT=\"a%2Fsin%28A%29=b%2Fsin%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%2F0.819152=b%2Fsin%2863.1%5Eo%29 ALIGN=MIDDLE  ALT=\"13%2F0.819152=b%2Fsin%2863.1%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%2F0.819152=b%2F0.891798 ALIGN=MIDDLE  ALT=\"13%2F0.819152=b%2F0.891798\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%2A0.891798%2F0.819152=b ALIGN=MIDDLE  ALT=\"13%2A0.891798%2F0.819152=b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=14.15 ALIGN=MIDDLE  ALT=\"b=14.15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "TRIANGLE #2:<BR>\n" );
document.write( "However, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=180%5Eo-61.9%5Eo=118.1%5Eo ALIGN=MIDDLE  ALT=\"C=180%5Eo-61.9%5Eo=118.1%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > also has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28C%29=0.882164 ALIGN=MIDDLE  ALT=\"sin%28C%29=0.882164\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and a triangle with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=55%5Eo ALIGN=MIDDLE  ALT=\"A=55%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=118.1%5Eo ALIGN=MIDDLE  ALT=\"C=118.1%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also possible,<BR>\n" );
document.write( "because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2BC=55%5Eo%2B118.1%5Eo ALIGN=MIDDLE  ALT=\"A%2BC=55%5Eo%2B118.1%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > is less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180%5Eo ALIGN=MIDDLE  ALT=\"180%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That triangle would have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=118.1%5Eo ALIGN=MIDDLE  ALT=\"C=118.1%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=180%5Eo-%28A%2BC%29=180%5Eo-%2855%5Eo%2B118.1%5Eo%29=180%5Eo-173.1%5Eo=6.9%5Eo ALIGN=MIDDLE  ALT=\"B=180%5Eo-%28A%2BC%29=180%5Eo-%2855%5Eo%2B118.1%5Eo%29=180%5Eo-173.1%5Eo=6.9%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=6.9%5Eo ALIGN=MIDDLE  ALT=\"B=6.9%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Fsin%28A%29=b%2Fsin%28B%29 ALIGN=MIDDLE  ALT=\"a%2Fsin%28A%29=b%2Fsin%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives us<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%2F0.819152=b%2Fsin%286.9%5Eo%29 ALIGN=MIDDLE  ALT=\"13%2F0.819152=b%2Fsin%286.9%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%2F0.819152=b%2F0.120137 ALIGN=MIDDLE  ALT=\"13%2F0.819152=b%2F0.120137\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%2A0.1201378%2F0.819152=b ALIGN=MIDDLE  ALT=\"13%2A0.1201378%2F0.819152=b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=1.91 ALIGN=MIDDLE  ALT=\"b=1.91\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );