SOLUTION: A quadrilateral PQRS has PQ = 18, QR = 50, RS = 45 and diagonal QS = 30. If QS bisects the angle at Q, what is the length of PS?

Algebra ->  Parallelograms -> SOLUTION: A quadrilateral PQRS has PQ = 18, QR = 50, RS = 45 and diagonal QS = 30. If QS bisects the angle at Q, what is the length of PS?      Log On


   

Question 1190326: A quadrilateral PQRS has PQ = 18, QR = 50, RS = 45 and diagonal QS = 30. If QS bisects the angle at Q, what is the length of PS?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let 2x be the measure of the angle that is bisected; so the measures of angles PQS and RQS are each x. Then

Using the law of cosines on triangle QRS,

45%5E2=50%5E2%2B30%5E2-2%2850%29%2830%29cos%28x%29 [1]

Using the law of cosines on triangle PQS,

x%5E2=18%5E2%2B30%5E2-2%2818%29%2830%29cos%28x%29 [2]

Solve [1] for cos(x) and substitute in [2].

I'll leave it to you to do the calculations....

If you only want the answer, without a chance to learn something from working on the problem, the answer is PS=27.