Answer:
The coordinates of point C are (7 , 6)
Step-by-step explanation:
* Lets explain how to solve the problem
- If point (x , y) divides a line segment whose endpoints are
[tex](x_{1},y_{1})[/tex] and [tex](x_{2},y_{2})[/tex] at ratio [tex]m_{1}:m_{2}[/tex]
from the first point, then [tex]x=\frac{x_{1}m_{2}+x_{2}m_{1}}{m_{1}+m_{2}}[/tex]
and [tex]y=\frac{y_{1}m_{2}+y_{2}m_{1}}{m_{1}+m_{2}}[/tex]
* Lets use this rule to solve the problem
- A, B, and C are col-linear, and B is between A and C
∴ A is [tex](x_{1},y_{1})[/tex]
∴ C is [tex](x_{2},y_{2})[/tex]
∴ B is (x , y)
- The ratio of AB to AC is 2 : 7
∵ AB : AC = 2 : 7
∴ AB is 2 parts of AC and BC is (7 - 2) = 5 parts of AC
∴ AB : BC = 2 : 5
∴ [tex]m_{1}:m_{2}=2:5[/tex]
∵ A = (0 , -8)
∴ [tex](x_{1},y_{1})[/tex] = (0 , -8)
∵ B = (2 , -4)
∴ (x , y) = (2 , -4)
∵ [tex]2=\frac{(0)(5)+(2)x_{2}}{2+5}[/tex]
∴ [tex]2=\frac{0+(2)x_{2}}{7}[/tex]
∴ [tex]2=\frac{(2)x_{2}}{7}[/tex]
- Multiply both sides by 7
∴ [tex]14=(2)x_{2}[/tex]
- Divide both sides by 2
∴ [tex]x_{2}=7[/tex]
* The x-coordinate of point C is 7
∵ [tex]-4=\frac{(-8)(5)+(2)y_{2}}{2+5}[/tex]
∴ [tex]-4=\frac{-40+(2)y_{2}}{7}[/tex]
- Multiply both sides by 7
∴ [tex]-28=-40+(2)y_{2}[/tex]
- Add 40 to both sides
∴ [tex]12=(2)y_{2}[/tex]
- Divide both sides by 2
∴ [tex]y_{2}=6[/tex]
* The y-coordinate of point C is 6
∴ The coordinates of point C are (7 , 6)
