document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117095>Question 1117095</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sides of a triangle are 50,60 and 70. Find the length of the angle bisector from the longest side to its opposite vertex.\r<BR>\n" );
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document.write( "Answer: 42.25\r<BR>\n" );
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document.write( "How to solve? </SPAN>\n" );
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document.write( "You have a triangle with a segment from one vertex to the opposite side; and you know -- or have enough information to determine -- the lengths of all but one of the segments in the figure.  In any problem like that, the use of Stewart's Theorem is a possible path to the solution.<br><BR>\n" );
document.write( "Let the triangle be ABC, and let AD be a segment with D on BC.  Then Stewart's Theorem says<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB%2ACD%2AAB%2BAC%2ABD%2AAC=AD%2ABC%2AAD%2BBD%2ADC%2ABC ALIGN=MIDDLE  ALT=\"AB%2ACD%2AAB%2BAC%2ABD%2AAC=AD%2ABC%2AAD%2BBD%2ADC%2ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Note the theorem holds for any such segment AD -- it can be a median, or an angle bisector, or an altitude; but it can also be ANY segment AD with D on BC.<br><BR>\n" );
document.write( "In your problem, AD is the angle bisector.  An angle bisector in a triangle divides the opposite side into two parts whose lengths are in the same ratio as the lengths of the two sides that form the angle.<br><BR>\n" );
document.write( "With AB = 50, AC = 60, and BC = 70, the angle bisector AD divides BC with length 70 into two pieces whose lengths are in the ratio 50:60, or 5:6.  That makes the lengths of the two segments BD = 350/11 and CD = 420/11.<br><BR>\n" );
document.write( "Then you are ready to plug in the segment lengths into the formula in Stewart's Theorem:<BR>\n" );
document.write( "AB=50; AC=60; BC=70; BD=350/11; CD=420/11;<BR>\n" );
document.write( "the unknown x is the length of AD<br><BR>\n" );
document.write( "Plugging in those values leads to the answer of 42.25 that you show for the length of AD.  I leave it to you to do the calculations. </SPAN>\n" );
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