document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207590>Question 1207590</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In triangle $PQR,$ let $X$ be the intersection of the angle bisector of $\angle P$ with side $QR$, and let $Y$ be the foot of the perpendicular from $X$ to side $PR$.  If $PQ = 10,$ $QR = 10,$ and $PR = 12,$ then compute the length of $XY$. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845582 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=845582\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "One response to your question has the right answer by a difficult path.  The other has the wrong solution because it applies the angle bisector theorem incorrectly.<br><BR>\n" );
document.write( "Here is a simpler correct solution.<br><BR>\n" );
document.write( "PQ=QR=10, so triangle PQR is isosceles with vertex Q.<br><BR>\n" );
document.write( "Let Z be the foot of the perpendicular from Q to side PR.  Since PQR is isosceles,  Z is the midpoint of PR, making ZR=6.<br><BR>\n" );
document.write( "Triangle QZR is a right triangle with leg ZR=6 and hypotenuse QR=10.  By the Pythagorean Theorem, that makes leg QZ=8.<br><BR>\n" );
document.write( "Triangles QZR and XYR are now similar. (Both are right triangles that have angle R in common.)<br><BR>\n" );
document.write( "PX is the bisector of angle P; by the angle bisector theorem, QX/RX = PQ/PR = 10/12 = 5/6.  That makes RX/RQ = 6/11.<br><BR>\n" );
document.write( "RX and RQ are the hypotenuses of similar triangles XYR and QZR.  Then, by corresponding parts of similar triangles, XY/QZ = 6/11.<br><BR>\n" );
document.write( "But QZ=8, making XY (6/11)*8 = 48/11.<br><BR>\n" );
document.write( "ANSWER: XZ = 48/11<br><BR>\n" );
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