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.

QTP and QTR are similar isosceles triangles.
Therefore,

  |PQ|     |QT|     |QT|        |QT|          |QT|
 ------ = ------ = ------ = ----------- = -----------
  |QT|     |QR|     |PS|     |PT|+|TS|     |PQ|+|TS|

              or

  |PQ|^2 + |TS||PQ| - |QT|^2 = 0

              or

  |PQ|^2 + 5|PQ| - 36 = 0

              or

  (|PQ|+9)(|PQ|-4) = 0

Thus, |PQ| = 4.

Posted by Bractals on 2007-05-01 12:50:29