A point S is taken on the side QR of triangle PQR such that RS = 2SQ. It is known that Angle PQR = 45^o and Angle SPQ = 15^o

Determine Angle PRQ.

We know the sum of angles in a triangle is 180°, therefore Angle PSQ = (180° - (45° + 15°)) = 120°.

Using the law of sines the length of PQ can be found:
  QS/sin(Angle QPS) = PQ/sin(Angle PSQ)
  QS/sin(15°) = PQ(sin(120°)
  (QS*(sin(120°))/sin(15°) = PQ
  3.346*QS ~= PQ

Using the law of cosines the length of PR can be found:
  PR² = PQ² + QR² - 2*PQ*QR*cos(Angle PQR)
  PR² ~= (3.346*QS)² + (3*QS)² - 2*(3.346*QS)*(3*QS)*cos(45°)
  PR² ~= 6*QS²
  PR ~= SQRT(6)*QS

Again, using the law of cosines, Angle PRQ can be found:
  PQ² = PR² + QR² - 2*PR*QR*cos(Angle PRQ)
  11.196*QS² ~= 6*QS² + 9*QS² - (14.7*QS²)*cos(Angle PRQ)
  (3.804/14.7) ~= cos(Angle PRQ)
  arccos(0.2588) ~= Angle PRQ
  Angle PRQ ~= 75°

Edited on October 30, 2007, 12:38 am

Posted by Dej Mar on 2007-10-30 00:06:39