SOLUTION: vt is an angle bisector of angle svu. If sv=9,vu=20,su=27, find st

In any triangle, the angle bisector divides the side to which it is drawn, in two segments proportional to the ratio 
of two other sides of a triangle.


    For this theorem and its proof, see the lesson
        On what segments the angle bisector divides the side of a triangle
    in this site.


Using this theorem for the triangle SVU, you get the proportion

 = ,   or    = .

The second equation is  |ST| + |TU| = 27.


Introducing for breavity x = |ST|, y = |TU|,  you have this system of 2 equations

20x - 9y = 0,       (1)
  x +  y = 27.      (2)


To solve it, use the Elimination method. For it, multiply eq(2) by 9 (both sides). Keep eq(1) as is. You will get

20x -  9y =   0,    (1)
 9x +  9y = 243.    (2)


Next add equations (1) and (2). You will get

29x = 243.


Hence,  x = |ST| = .