Consider the parallelogram PQRS with PS parallel to QR and PQ parallel to SR. The bisector of the angle PQR intersects PS at T.

Determine PQ, given that TS = 5, QT = 6 and RT = 6.

By definition, angles PQT, RQT, and PTQ are all equal.  This makes triangle PTQ an isosceles triangle.  Triangle QTR by definition is also isosceles.  It is also similar to triangle PTQ since their angles are the same.  This means their sides are proportional.

Let's define the sides of the parallelogram as A = PQ, B = QR.  The ratios of the sides of the similar triangles gives the following equivalency:

A/6 = 6/B       or      AB = 36

Since PT = PQ = A, the measurement of QR = B = A + 5.

There are now two equations and two unknowns.  Substituting for B:

A(A+5) = 36

AČ + 5A - 36 = 0

(A+9)(A-4) = 0

A = 4, B = 9

PQ = 4

Posted by hoodat on 2007-05-01 15:57:46