document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=819293>Question 819293</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A triangle ABC has sides with lengths a= 12 cm, b = 8 cm, and the angle B = 30º. What are the possible values for the length c of the third side of the triangle? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #493080 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=493080\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C200%2C-2%2C18%2C-2%2C8%2C%0D%0Atriangle%280%2C0%2C10.39%2C0%2C10.39%2C6%29%2C%0D%0Arectangle%2810.39%2C0%2C10.09%2C0.3%29%2C%0D%0Atriangle%28-3%2C0%2C-3%2C30%2C30%2C0%29%2C%0D%0Alocate%28-0.1%2C-0.1%2CB%29%2Clocate%2810.3%2C6.7%2CC%29%2C%0D%0Alocate%2810.3%2C-0.1%2CD%29%2Clocate%281.1%2C1.1%2C30%5Eo%29%2C%0D%0Ared%28arc%280%2C0%2C5.5%2C5.5%2C330%2C360%29%29%2C%0D%0Alocate%2810.5%2C3.5%2C6%29%2Clocate%285%2C2.88%2C12%29%2C%0D%0Ablue%28line%2815.68%2C0%2C10.39%2C6%29%29%2Clocate%2812.7%2C3%2Cblue%288%29%29%2C%0D%0Agreen%28line%285.1%2C0%2C10.39%2C6%29%29%2Clocate%287.7%2C3%2Cgreen%288%29%29%2C%0D%0Alocate%2815.58%2C0%2Cblue%28Y%29%29%2Clocate%285%2C0%2Cgreen%28X%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ABC could be XBC OR YBC. vertex A could be X or Y, then side b (AB) would be XC or YC, measuring 8cm.<BR>\n" );
document.write( "Since there were two possible triangles, I drew a third triangle, right triangle BCD.<BR>\n" );
document.write( "CD = BC<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%2830%5Eo%29=12%2A0.5=6 ALIGN=MIDDLE  ALT=\"sin%2830%5Eo%29=12%2A0.5=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can also calculate<BR>\n" );
document.write( "BD = BC<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%2830%5Eo%29=12%2A%28sqrt%283%29%2F2%29=6sqrt%283%29=about10.39 ALIGN=MIDDLE  ALT=\"cos%2830%5Eo%29=12%2A%28sqrt%283%29%2F2%29=6sqrt%283%29=about10.39\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can calculate the length of XD = YD in congruent right triangles XDC and YDC.<BR>\n" );
document.write( "That length is<BR>\n" );
document.write( "XD = YD =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288%5E2-6%5E2%29sqrt%2864-36%29=sqrt%2828%29=2sqrt%287%29=about5.29 ALIGN=MIDDLE  ALT=\"sqrt%288%5E2-6%5E2%29sqrt%2864-36%29=sqrt%2828%29=2sqrt%287%29=about5.29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "BX = BD - XD =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%286sqrt%283%29-2sqrt%287%29%29 ALIGN=MIDDLE  ALT=\"highlight%286sqrt%283%29-2sqrt%287%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= about<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%285.10%29 ALIGN=MIDDLE  ALT=\"highlight%285.10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "BY = BD + YD =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%286sqrt%283%29%2B2sqrt%287%29%29 ALIGN=MIDDLE  ALT=\"highlight%286sqrt%283%29%2B2sqrt%287%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= about<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2815.68%29 ALIGN=MIDDLE  ALT=\"highlight%2815.68%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the approximate measure of the third side is<BR>\n" );
document.write( "either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%285.10cm%29 ALIGN=MIDDLE  ALT=\"highlight%285.10cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2815.68cm%29 ALIGN=MIDDLE  ALT=\"highlight%2815.68cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "ALTERNATE SOLUTION:<BR>\n" );
document.write( "Maybe your teacher expected you to use law of cosines,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=a%5E2%2Bc%5E2-2ac%2Acos%28B%29 ALIGN=MIDDLE  ALT=\"b%5E2=a%5E2%2Bc%5E2-2ac%2Acos%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%5E2=12%5E2%2Bc%5E2-2%2A12%2Ac%2Acos%2830%5Eo%29 ALIGN=MIDDLE  ALT=\"8%5E2=12%5E2%2Bc%5E2-2%2A12%2Ac%2Acos%2830%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=64=144%2Bc%5E2-2%2A12%2Ac%2A%28sqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"64=144%2Bc%5E2-2%2A12%2Ac%2A%28sqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64=144%2Bc%5E2-%2812sqrt%283%29%29c ALIGN=MIDDLE  ALT=\"64=144%2Bc%5E2-%2812sqrt%283%29%29c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2-%2812sqrt%283%29%29c%2B144-64=0 ALIGN=MIDDLE  ALT=\"c%5E2-%2812sqrt%283%29%29c%2B144-64=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2-%2812sqrt%283%29%29c%2B80=0 ALIGN=MIDDLE  ALT=\"c%5E2-%2812sqrt%283%29%29c%2B80=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That quadratic equation can be solved using the quadratic formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+%2812sqrt%283%29+%2B-+sqrt%28%2812sqrt%283%29%29%5E2-4%2A1%2A80+%29%29%2F%282%2A1%29=%2812sqrt%283%29+%2B-+sqrt%2812%5E2%2A3-320+%29%29%2F2=%2812sqrt%283%29+%2B-+sqrt%28432-320%29%29%2F2=%2812sqrt%283%29+%2B-+sqrt%28112%29%29%2F2=%2812sqrt%283%29+%2B-+sqrt%2816%2A7%29%29%2F2=%2812sqrt%283%29+%2B-+4sqrt%287%29%29%2F2=highlight%286sqrt%283%29+%2B-+2sqrt%287%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The quadratic equation can also be solved by completing the square:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2-%2812sqrt%283%29%29c%2B80=0 ALIGN=MIDDLE  ALT=\"c%5E2-%2812sqrt%283%29%29c%2B80=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2-%2812sqrt%283%29%29c=-80 ALIGN=MIDDLE  ALT=\"c%5E2-%2812sqrt%283%29%29c=-80\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2-%2812sqrt%283%29%29c%2B%286sqrt%283%29%29%5E2=%286sqrt%283%29%29%5E2-80 ALIGN=MIDDLE  ALT=\"c%5E2-%2812sqrt%283%29%29c%2B%286sqrt%283%29%29%5E2=%286sqrt%283%29%29%5E2-80\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28c-6sqrt%283%29%29%5E2=6%5E2%2A3-80 ALIGN=MIDDLE  ALT=\"%28c-6sqrt%283%29%29%5E2=6%5E2%2A3-80\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28c-6sqrt%283%29%29%5E2=108-80 ALIGN=MIDDLE  ALT=\"%28c-6sqrt%283%29%29%5E2=108-80\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28c-6sqrt%283%29%29%5E2=28 ALIGN=MIDDLE  ALT=\"%28c-6sqrt%283%29%29%5E2=28\"   DISABLED_style= \"max-width:90%; height:auto;\" > so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28c-6sqrt%283%29=sqrt%2828%29=2sqrt%287%29%2C+%22OR%22%2Cc-6sqrt%283%29=sqrt%2828%29=-2sqrt%287%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "leading to the solutions <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28c=6sqrt%283%29+%2B-+2sqrt%287%29%29 ALIGN=MIDDLE  ALT=\"highlight%28c=6sqrt%283%29+%2B-+2sqrt%287%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "ANOTHER ALTERNATE:<BR>\n" );
document.write( "Since you have the measures of angle B and side b, you can apply law of sines, and find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28A%29 ALIGN=MIDDLE  ALT=\"sin%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and two possible approximate measures for angle A.<BR>\n" );
document.write( "Then you could calculate the approximate measures for the two options for angle C, and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28C%29 ALIGN=MIDDLE  ALT=\"sin%28C%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and then use law of sines again to find the two possible measures for side c. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );