Answer to Question #95302 in Geometry for Emmanuel

Question

In a pparallelogramABCD, P divides AB in the ratio 2 : 5 and Q divides DC in the ratio 3 : 2. If AC and PQ intersect at R, find the ratios AR : RC and PR : RQ.

Answer 1

 
Solution:
Consider triangles APR and RCQ
angle PAR = angle QCR with parallel lines AD and BC
angle APR = angle RQC with parallel lines AB and CD
angle PRA = angle CRQ as vertical angles
So the triangles APR and RCQ are similar because the three corresponding angles are equal 
We write the equation for finding the side AB:
AP + PB = AB
2х + 5х = AB
We write the equation for finding the side CD:
CQ + QD = CD
2у + 3у = CD
The sides of the parallelogram are equal CD=AB, therefore, we have
AP + PB = CQ + QD
2х + 5х = 2у + 3у 
7х = 5у
х : у = 5 : 7
Record the corresponding ratios of the sides in triangles APR and RCQ:
AP : CQ = AR : RC = PR : RQ (1)
AP : CQ = 2x : 2y = x : y =5 : 7 (2)
From formulas (1), (2), it follows that
AR : RC = 5 : 7
PR : RQ = 5 : 7

Answer:AR : RC = 5 : 7
       PR : RQ = 5 : 7