document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=26185>Question 26185</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a landscape architect designed a flower bed in a shape of a triangle with side lengths 12,14,15.the architech needs to know the area of the flower bed.\r<BR>\n" );
document.write( "\n" );
document.write( "My Notes to you: it shows a triangle. base is 14 and the sides are 13 and 15. with a line down the middle.\r<BR>\n" );
document.write( "\n" );
document.write( "I belive Im suppose to use the pangerium therom \r<BR>\n" );
document.write( "\n" );
document.write( "All I need to know how to find the height.\r<BR>\n" );
document.write( "\n" );
document.write( "Ive tried to split 14 in different ways but the height is not the same on both sides\r<BR>\n" );
document.write( "\n" );
document.write( "Thanks for all the help!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #14161 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=14161\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Well, you must be studying a new theorem because I've never heard of the \"Pangerium theorem\"! Perhaps you meant to write \"Pythagorean theorem\" but if the problem is to find the area of the triangle, you don't need to find the height since you are given the lengths of the three sides.  You can use Heron's formula for finding the area if you know the lengths of the three sides. a = 13, b = 14, and c = 15<BR>\n" );
document.write( "Heron's formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+sqrt%28S%28S-a%29%28S-b%29%28S-c%29%29 ALIGN=MIDDLE  ALT=\"A+=+sqrt%28S%28S-a%29%28S-b%29%28S-c%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Where: S is the semi-perimeter of the triangle and a, b, and c are the lengths of the sides.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%281%2F2%29%28a%2Bb%2Bc%29 ALIGN=MIDDLE  ALT=\"S+=+%281%2F2%29%28a%2Bb%2Bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%281%2F2%29%2813%2B14%2B15%29 ALIGN=MIDDLE  ALT=\"S+=+%281%2F2%29%2813%2B14%2B15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%281%2F2%29%2842%29 ALIGN=MIDDLE  ALT=\"S+=+%281%2F2%29%2842%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+21 ALIGN=MIDDLE  ALT=\"S+=+21\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+sqrt%2821%2821-13%29%2821-14%29%2821-15%29%29 ALIGN=MIDDLE  ALT=\"A+=+sqrt%2821%2821-13%29%2821-14%29%2821-15%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+sqrt%2821%288%29%287%29%286%29%29 ALIGN=MIDDLE  ALT=\"A+=+sqrt%2821%288%29%287%29%286%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+sqrt%287056%29 ALIGN=MIDDLE  ALT=\"A+=+sqrt%287056%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+84 ALIGN=MIDDLE  ALT=\"A+=+84\"   DISABLED_style= \"max-width:90%; height:auto;\" >sq.units.\r<BR>\n" );
document.write( "\n" );
document.write( "But now, let's try to find the height of the scalene triangle.  The base is 14, so when we drop a perpendicular line from the vertex angle to the base, the base is split into two unequal parts.<BR>\n" );
document.write( "Let's label one part x. The other part will then be 14-x. So we now have two right triangles whose dimension are:<BR>\n" );
document.write( "1) Hypotenuse = 13, base = x, and the height = h.<BR>\n" );
document.write( "2) Hypotenuse = 15, the base = 14-x, and the height = h (same as the h above)\r<BR>\n" );
document.write( "\n" );
document.write( "Now we can use the Pythgorean theorem <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2+=+a%5E2+%2B+b%5E2 ALIGN=MIDDLE  ALT=\"c%5E2+=+a%5E2+%2B+b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >to find the height, h.\r<BR>\n" );
document.write( "\n" );
document.write( "For triangle 1):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%5E2+=+h%5E2+%2B+x%5E2 ALIGN=MIDDLE  ALT=\"13%5E2+=+h%5E2+%2B+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve this for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2 ALIGN=MIDDLE  ALT=\"h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+13%5E2+-+x%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+=+13%5E2+-+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+169+-+x%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+=+169+-+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "For triangle 2):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%5E2+=+h%5E2+%2B+%2814-x%29%5E2 ALIGN=MIDDLE  ALT=\"15%5E2+=+h%5E2+%2B+%2814-x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve this for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2 ALIGN=MIDDLE  ALT=\"h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+15%5E2+-+%2814-x%29%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+=+15%5E2+-+%2814-x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+225+-+%28196+-28x+%2B+x%5E2%29 ALIGN=MIDDLE  ALT=\"h%5E2+=+225+-+%28196+-28x+%2B+x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+29+%2B+28x+-+x%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+=+29+%2B+28x+-+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+h%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+=+h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=169+-+x%5E2+=+29+%2B+28x+-+x%5E2 ALIGN=MIDDLE  ALT=\"169+-+x%5E2+=+29+%2B+28x+-+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify and solve for x. Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=169+=+29+%2B+28x ALIGN=MIDDLE  ALT=\"169+=+29+%2B+28x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 29 from both sides.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=140+=+28x ALIGN=MIDDLE  ALT=\"140+=+28x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 28.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5+=+x ALIGN=MIDDLE  ALT=\"5+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5 ALIGN=MIDDLE  ALT=\"x+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now substitute this value into one of the two equations for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2 ALIGN=MIDDLE  ALT=\"h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and solve for h.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+169+-+x%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+=+169+-+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+169+-+5%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+=+169+-+5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+169+-+25 ALIGN=MIDDLE  ALT=\"h%5E2+=+169+-+25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+144 ALIGN=MIDDLE  ALT=\"h%5E2+=+144\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of bothsides.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+12 ALIGN=MIDDLE  ALT=\"h+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Discard the negative solution as the height can only be positive.\r<BR>\n" );
document.write( "\n" );
document.write( "So, finally, we have the height of the scalene triangle and now we can find the area using: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+%281%2F2%29bh ALIGN=MIDDLE  ALT=\"A+=+%281%2F2%29bh\"   DISABLED_style= \"max-width:90%; height:auto;\" > where b = 14 and h = 12.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+%281%2F2%29%2814%29%2812%29 ALIGN=MIDDLE  ALT=\"A+=+%281%2F2%29%2814%29%2812%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+84 ALIGN=MIDDLE  ALT=\"A+=+84\"   DISABLED_style= \"max-width:90%; height:auto;\" > sq. units.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );