document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=874948>Question 874948</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how to find the area of the triangle whose sides are 32m, 38m, and 45m? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #527856 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=527856\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There is usually more than one way to solve a problem, and in this case, I immediately think of two ways. If this is a homework problem, you would want to use the one your teacher expects, based on what you have been taught in class.<BR>\n" );
document.write( "Otherwise, you would use the one that is most efficient for you.<BR>\n" );
document.write( " <BR>\n" );
document.write( "USING HERON'S FORMULA:<BR>\n" );
document.write( "Heron's formula can be written in many ways.<BR>\n" );
document.write( "Here is one of them.<BR>\n" );
document.write( "If the lengths of the sides of a triangle (in whatever units) are<BR>\n" );
document.write( "<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=b ALIGN=MIDDLE  ALT=\"b\"   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;\" > ,<BR>\n" );
document.write( "and we define the semi-perimeter as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=%28a%2Bb%2Bc%29%2F2 ALIGN=MIDDLE  ALT=\"s=%28a%2Bb%2Bc%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "then the area of the triangle (in whatever units squared) can be calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28s%28s-a%29%28s-b%29%28s-c%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28s%28s-a%29%28s-b%29%28s-c%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In your case, the lengths in meters are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=32 ALIGN=MIDDLE  ALT=\"a=32\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=38 ALIGN=MIDDLE  ALT=\"b=38\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=45 ALIGN=MIDDLE  ALT=\"c=45\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=%2832%2B38%2B45%29=115%2F2=57.5 ALIGN=MIDDLE  ALT=\"s=%2832%2B38%2B45%29=115%2F2=57.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the area, in square meters can be calculated as either<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28115%2F2%29%28115%2F2-32%29%28115%2F2-38%29%28115%2F2-45%29%29=sqrt%28%28115%2F2%29%2A%2851%2F2%29%2A%2839%2F2%29%2A%2825%2F16%29%29=sqrt%285718375%29%2F4=highlight%2815sqrt%2825415%29%2F4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2857.5%2857.5-32%29%2857.5-38%29%2857.5-45%29%29=sqrt%2857.5%2A25.5%2A19.5%2A12.5%29=highlight%28sqrt%28357398.4375%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Either way, that is approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28597.8%29 ALIGN=MIDDLE  ALT=\"highlight%28597.8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If we wanted an exact expression,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2815sqrt%2825415%29%2F4%29 ALIGN=MIDDLE  ALT=\"highlight%2815sqrt%2825415%29%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the most elegant way to write it.<BR>\n" );
document.write( " <BR>\n" );
document.write( "USING LAW OF COSINES:<BR>\n" );
document.write( "With two sides, and the sine of the angle in between,<BR>\n" );
document.write( "we can calculate the area.<BR>\n" );
document.write( "Having the length of all 3 sides,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=32 ALIGN=MIDDLE  ALT=\"a=32\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=38 ALIGN=MIDDLE  ALT=\"b=38\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=45 ALIGN=MIDDLE  ALT=\"c=45\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the law of cosines allows us to calculate the measure of any one of the angles.<BR>\n" );
document.write( "With two sides, and the angle in between, <BR>\n" );
document.write( "If we wanted to calculate the measure of <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( "the angle opposite side <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=38 ALIGN=MIDDLE  ALT=\"b=38\"   DISABLED_style= \"max-width:90%; height:auto;\" > and between the other two sides,<BR>\n" );
document.write( "we would use<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( "Substituting the values for <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=b ALIGN=MIDDLE  ALT=\"b\"   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 get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=38%5E2=32%5E2%2B45%5E2-2%2A32%2A45%2Acos%28B%29 ALIGN=MIDDLE  ALT=\"38%5E2=32%5E2%2B45%5E2-2%2A32%2A45%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=1444=1024%2B2025-2880cos%28B%29 ALIGN=MIDDLE  ALT=\"1444=1024%2B2025-2880cos%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=1444=3049-2880cos%28B%29 ALIGN=MIDDLE  ALT=\"1444=3049-2880cos%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=2880cos%28B%29=3049-1444 ALIGN=MIDDLE  ALT=\"2880cos%28B%29=3049-1444\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2880cos%28B%29=1605 ALIGN=MIDDLE  ALT=\"2880cos%28B%29=1605\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28B%29=1605%2F2880 ALIGN=MIDDLE  ALT=\"cos%28B%29=1605%2F2880\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28B%29=107%2F192 ALIGN=MIDDLE  ALT=\"cos%28B%29=107%2F192\"   DISABLED_style= \"max-width:90%; height:auto;\" > (simplifying).<BR>\n" );
document.write( "We do not need to calculate <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( "because we can calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28B%29 ALIGN=MIDDLE  ALT=\"sin%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > directly from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28B%29 ALIGN=MIDDLE  ALT=\"cos%28B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28B%29=sqrt%281-%28cos%28B%29%29%5E2%29 ALIGN=MIDDLE  ALT=\"sin%28B%29=sqrt%281-%28cos%28B%29%29%5E2%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%28B%29=sqrt%281-%28107%2F192%29%5E2%29=sqrt%28192%5E2%2F192%5E2-107%5E2%2F192%5E2%29=sqrt%28%28192%5E2-107%5E2%29%2F192%5E2%29=sqrt%28192%5E2-107%5E2%29%2F192=sqrt%2825415%29%2F192 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So we can calculate the area of the triangle as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2Aa%2Ac%2Asin%28B%29=%281%2F2%29%2A32%2A45%2A%28sqrt%2825415%29%2F192%29=720%2A%28sqrt%2825415%29%2F192%29=720sqrt%2825415%29%2F192=highlight%2815sqrt%2825415%29%2F4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As an approximate value, that is approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28597.8%29 ALIGN=MIDDLE  ALT=\"highlight%28597.8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );