Question 1090433
Q is the midpoint of PT,
so PQ=QT.
R is the midpoint of QT,
so QR=RT.
S is the midpoint off RT,
so RS=ST.
Let us say RS=ST=x .
Then, RT=RS+ST=x+x=2x ,
so RT=QR=2x .
Then, QT=QR+RT=2x+2x=4x ,
and PQ=QT=4x .
Then, PS=PQ+QR+RS=4x+2x+x=7x ,
and PT=PQ+QT=4x+4x=8x .
The problem says that PS=9 ,
so {{{7x=9}}} , {{{x=9/7}}} ,
and PT=8x={{{8*(9/7)=highlight(72/7=10&2/7)}}} .
 
{{{drawing(550,100,-0.3,10.7,-1,1,
line(-0.3,0,10.7,0),
line(0,-0.2,0,0.2),line(1,-0.2,1,0.2),
line(3,-0.2,3,0.2),line(4,-0.2,4,0.2),
line(5,-0.2,5,0.2),line(6,-0.2,6,0.2),
line(7,-0.2,7,0.2),line(8,0.2,8,-0.2),
line(2,-0.2,2,0.2),line(9,-0.2,9,0.2),
line(10,-0.2,10,0.2),circle(0,0,0.1),
circle(36/7,0,0.1),circle(54/7,0,0.1),
circle(63/7,0,0.1),circle(72/7,0,0.1),
locate(-0.1,-0.2,0),locate(4.9,-0.2,5),
locate(9.8,-0.2,10),locate(-0.1,0.5,P),
locate(5.04,0.5,Q),locate(7.6,0.5,R),
locate(8.9,0.5,S),locate(10.2,0.5,T)
)}}}