SOLUTION: In triangle RST, U lies on TS with TU:US= 2:3. M is the midpoint of RU. TM intersects RS in V. Find the ratio of RV:RS. I was able to solve it with trig, and I got 2:7. It was me

Question 1074278: In triangle RST, U lies on TS with TU:US= 2:3. M is the midpoint of RU. TM intersects RS in V. Find the ratio of RV:RS.
I was able to solve it with trig, and I got 2:7. It was meant to be solved with basic geometry, but I haven't be able to think of a way.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Maybe by now you have been able to think of a way.
I could not think of a way late last night,
but I woke up with a clearer mind this morning,
so I added line MW, parallel to ST:
MW is part of two new triangles similar to two triangles that were there before.

Now that I look at the drawing,
I realize it has TU:RS = 3:2 .
A drawing is not necessary to support the reasoning,
but even an inaccurate drawing helps
as a reminder of the meaning of each point.

RWM is similar to RSU.
MVW is similar to TVS.
The "scale down factors" between the similar triangles
lead to a way to calculate the lengths of RW and VW
as fractions of the length pf RS,
and that allows to calculate, as a fraction of the length of RS,
the lengths of .

Since and ,

So,
and .
Since M is the midpoint of RU,
and WM is parallel to ST,
RWM is not only similar to RSU;
it is a scale version of RSU.
So, not only is MW the midsegment of RSU,
with ,
but also ,
and of course, too.

.
Since MVW is similar to TVS,
and , .

Finally,
--> .
Form there,
--> --> --> ,
and --> -->

NOTE:
I am not sure if the reasoning above is the shortest way to the answer.
Maybe I should have drawn a line through M parallel to RS instead.
With nothing else to do, I would explore other angles to this problem
(and draw a better drawing).
However, if you are still looking for a geometry-based answer,
a less than perfect answer is better than no answer.
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