ΔRST, U lies on TS with TU:US = 2:3. M is the midpoint of RU. TM intersects RS at V. Find RV:RS.

  2 : 7

Step-by-step explanation:

Draw a line parallel to RS through U. Label its intersection with TM as point X. This will create triangle UMX congruent to triangle RMV by the ASA postulate. (Vertical angles at M are congruent; MR = MU by definition: and angles VRM and XUM are alternate interior angles at a transversal of parallel lines.) Then UX = RV by CPCTC.

The line UX has also created similar triangles TUX and TSV. Their scale factor is TU : TS = 2 : (2+3) = 2 : 5. The sides UX : SV have the same scale factor, 2 : 5.

From above, we know that UX = RV, which tells us the scale factor of RV : SV is 2 : 5. Thus ...

  RV:RS = RV:(RV+SV) = 2:(2+5)

  RV:RS = 2:7

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Comment on the solution

I had to draw the attached figure to see what the ratio was. Then I had to figure out a way to show why. The congruent angles at M and the equal sides RM and UM offered a clue as to how to do it.

add an image, and i will answer

step-by-step explanation: