The cordinates of the endpoints of RT are R(-6,-5) and T(4,0), and point S is on RT. The coordinates of S are (-2,-3). Which of the following represent the ratio RS:ST?

The ratio RS:ST is 2:3.

We will use the distance formula:
[tex]d=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]

The distance from R to S is:
[tex]d=\sqrt{(-5--3)^2+(-6--2)^2} \\ \\=\sqrt{(-5+3)^2+(-6+2)^2} \\ \\=\sqrt{(-2)^2+(-4)^2} \\ \\=\sqrt{4+16}=\sqrt{20}=\sqrt{4*5}=2\sqrt{5}[/tex]

The distance from S to T is:
[tex]d=\sqrt{(-3-0)^2+(-2-4)^2} \\ \\=\sqrt{(-3)^2+(-6)^2}=\sqrt{9+36}=\sqrt{45} \\ \\=\sqrt{9*5}=3\sqrt{5}[/tex]

The ratio of RS to ST is then
[tex]\frac{2\sqrt{5}}{3\sqrt{5}}[/tex]

Since √5 cancels out on the top and bottom, we are left with 
2/3 = 2:3