document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1143115>Question 1143115</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Three sides of a triangle measures 12 cm, 18 cm and 10 cm. Find the altitude of the longest side, the median of the longest side, the length of angle bisector of the largest angle and the line segment joining the midpoints of 12 cm and 18 cm sides. Thanks.<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 #763903 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=763903\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "(1) length of the altitude to the longest side...<br><BR>\n" );
document.write( "As shown by tutor @ikleyn, the area of the triangle, from Heron's formula, is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40sqrt%282%29 ALIGN=MIDDLE  ALT=\"40sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Given that area and the length of the base 18, the height is calculated from the most common formula for the area of a triangle:<BR>\n" );
document.write( "<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;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40sqrt%282%29+=+%281%2F2%29%2818%29h ALIGN=MIDDLE  ALT=\"40sqrt%282%29+=+%281%2F2%29%2818%29h\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+%2840%2F9%29sqrt%282%29 ALIGN=MIDDLE  ALT=\"h+=+%2840%2F9%29sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Use Stewart's Theorem to find the lengths of the median to the longest side and the length of the bisector of the largest angle.<br><BR>\n" );
document.write( "Here is an ugly text description....<br><BR>\n" );
document.write( "(See the wikipedia article on Stewart's Theorem for a diagram)<br><BR>\n" );
document.write( "Let the base of the triangle be BC (length a).  Then side AC has length b and side AB has length c.  P is any point on base BC; d is the length of AP.  Let CP=n and BP=m.  Then the theorem relates the lengths of all the segments as follows:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=bmb%2Bcnc+=+dad%2Bamn ALIGN=MIDDLE  ALT=\"bmb%2Bcnc+=+dad%2Bamn\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Note the formula is usually written using b^2, c^2, and d^2.  I like the format I showed because, in applying the theorem, it gives you a visual picture of the relationship between the calculations \"bmb\", \"cnc\" and \"dad\".<br><BR>\n" );
document.write( "(2) length of the median to the longest side...<br><BR>\n" );
document.write( "You know m=n=9 because AP is the median.  The length of the median, d, is the only unknown.  Plug in the known values and solve for d.<br><BR>\n" );
document.write( "(3) length of the angle bisector of the largest angle...<br><BR>\n" );
document.write( "The angle bisector divides the base 18 into two pieces whose lengths are in the same ratio as the lengths of the two sides that form the angle.  Divide 18 into two parts in the ratio 10:12 to find the values of m and n; then perform the same calculation as above with the new values for m and n.<br><BR>\n" );
document.write( "(4) length of the segment joining the midpoints of the 12cm and 18cm sides....<br><BR>\n" );
document.write( "This is by far the easiest of the problems -- it is half the length of the 10cm side. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );