Ration of segments question?

Question

1. The problem statement, all variables and given/known data

For triangle XYZ, point P divides XZ in the ratio 3:1 and Q is the midpoint of XY. If R is the point of intersection of PY and QZ, find the ratio into which R divides PY.

2. Relevant equations

This is the only equation that may pertain to this that I can think of.
For line segment APB, vector OP= b/(a+b) OA + a/(a+b) OB, where O is any point and and b are the ratios.


3. The attempt at a solution

I really need help, this is all i can come up with.

we are looking for PR:RY

RP=1/4 RX + 3/4 RZ
RQ=1/2 RX + 1/2 RY

and RP, RZ, RQ, AND RY are vectors

help please

Answer 1

This is a great problem. I did one like this this morning. The strategey is to extend ray XZ to a point T such that XZ is congruent to ZT. Draw YT. Now in triangle XYT, you have the midline QZ. Therefore QZ is parallel to YT.

Now look at triangle PYT. RZ is parallel to YT, so it cuts the sides proportionally. PZ: ZT is 1:4, hence PR:RY is likewise.

Similar problem is at:
http://answers.yahoo.com/question/index;_ylt=AvHYRuUkDeKJ5.HmaTOndpTsy6IX?qid=20070104103415AAvnoau

Answer 2

1. The problem statement, all variables and given/known data

For triangle XYZ, point P divides XZ in the ratio 3:1 and Q is the midpoint of XY. If R is the point of intersection of PY and QZ, find the ratio into which R divides PY.

I believe the ratio is 4:1.