ANSWER
[tex]3: 2[/tex]
EXPLANATION
The equation of the line connecting (3,-1) and (8,9) is
[tex]y - 2x = - 7...(1)[/tex]
The given line segment also have equation
[tex]x - y - 2 = 0[/tex]
Or
[tex]x - y = 2..(2)[/tex]
Adding equation (2) and (1) gives,
[tex] - x = - 5[/tex]
[tex]x = 5[/tex]
We substitute x=5 into equation (2) to get,
[tex]5 - y = 2[/tex]
[tex]y = 3[/tex]
This means that the point of intersection is (5,3)
Let this point divide the line segment joining (3, -1) and (8, 9) in the ratio
m:n
Then,
[tex] \frac{3m + 8n}{m + n} = 5[/tex]
This implies that,
[tex]5m + 5n = 3m + 8n[/tex]
[tex]2m = 3n[/tex]
[tex] \frac{m}{n} = \frac{3}{2} [/tex]
m:n=3:2
