document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193785>Question 1193785</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given:	isosceles trapezoid MNPQ with QP = 18 and m∠M = 120°<BR>\n" );
document.write( "the bisectors of ∠s MQP and NPQ meet at point T on MN<BR>\n" );
document.write( "Find:	the perimeter of MNPQ </SPAN>\n" );
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document.write( "Here is a diagram with some lines added...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28800%2C400%2C-10%2C10%2C-2%2C8%0D%0A%2Cline%28-9%2C0%2C9%2C0%29%2Cline%28-6%2C3sqrt%283%29%2C6%2C3sqrt%283%29%29%2Cline%28-9%2C0%2C-6%2C3sqrt%283%29%29%2Cline%289%2C0%2C6%2C3sqrt%283%29%29%2Cline%28-9%2C0%2C0%2C3sqrt%283%29%29%2Cline%289%2C0%2C0%2C3sqrt%283%29%29%0D%0A%2Cline%280%2C0%2C0%2C3sqrt%283%29%29%2Cline%289%2F2%2C3sqrt%283%29%2F2%2C6%2C3sqrt%283%29%29%0D%0A%2Clocate%28-6%2C6%2CM%29%2Clocate%286%2C6%2CN%29%2Clocate%289%2C-0.5%2CP%29%2Clocate%28-9%2C-0.5%2CQ%29%0D%0A%2Clocate%280%2C6%2CT%29%2Clocate%280%2C-0.5%2CU%29%2Clocate%284%2C2.5%2CV%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "QT bisects angle MQP and PT bisects angle NPQ; angles MQP and NPQ are each 60 degrees, so angles PQT and TPQ are each 30 degrees.  So triangle PTQ is isosceles, with base angles 30 degrees.<br><BR>\n" );
document.write( "TU is the altitude of triangle PTQ; it divides triangle PTQ into congruent 30-60-90 right triangles PTU and QTU.  Since QP is 18, PU and QU are each 9.<br><BR>\n" );
document.write( "PU with length 9 is the long leg of a 30-60-90 right triangle, so TU is 9/sqrt(3) = 3*sqrt(3), and PT is twice TU or 6*sqrt(3).<br><BR>\n" );
document.write( "Angle MNP is 120 degrees, so angle NPQ is 60 degrees; PT bisects angle NPQ, so angle NPT is 30 degrees; since angle MNP is 120 degrees, triangle TNP is also isosceles with base angles 30 degrees.<br><BR>\n" );
document.write( "Altitude NV of triangle TNP divides triangle TNP into two congruent 30-60-90 right triangles, each with long leg equal to half of TP, which is 3*sqrt(3).  So in triangle TVN NV is 3 and TN is 6.<br><BR>\n" );
document.write( "Then triangles QMT and PNT are both isosceles with legs of length 6, and the perimeter of trapezoid MNPQ is 6+6+9+9+6+6 = 42.<br><BR>\n" );
document.write( "ANSWER: 42<br><BR>\n" );
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