Question

In the triangle given below OP= p and OQ = q.R is a point on PQ  such that PR:RQ = 1 : 3 and that 5OS= 2OQ.PS and OR intersect at T.

1. Express in terms of p and q.
   • OS
   • PQ
   • OR
2. Given that OT=hOR and PT = kPS, determine the values of h and k. 

Answer 1

1.    
   • OS    2/5q
   • PQ    q-p
   • OR    3/4p+1/4q
2.    
   OT=h (³/₄p + ¹/₄q)
   OT=p +k (²/₅ q – p)
   ³/₄hp + ¹/₄hq = ¹/₄hq=p+²/₅kq –kp
   ³/₄h = 1-k ^h/₄=²/₅k
   ³/₄h (^8k/₅) (1-k)
   6^k/₅ = 5-5k
   11k=5
   K=⁵/₁₁
   11=⁸/₁₁
   PT:T=⁵/₁₁:⁶/₁₁
   5:6