Respuesta :
Answer:
C(-3,3)
Step-by-step explanation:
Given
A = (7,-1)
B = (2,1)
AB:AC = 1:2
Required
Determine the coordinates of C
Since, B is between A and C; we need to determine ratio BC as follows;
[tex]AB:AC = 1:2[/tex]
Convert to division
[tex]\frac{AB}{AC} = \frac{1}{2}[/tex]
AC = AB + BC;
[tex]\frac{AB}{AB + BC} = \frac{1}{2}[/tex]
Cross Multiply
[tex]2 * AB = 1 * (AB + BC)[/tex]
[tex]2 AB = AB + BC[/tex]
[tex]2AB - AB = BC[/tex]
[tex]AB = BC[/tex]
Divide both sides by BC
[tex]\frac{AB}{BC} = 1[/tex]
Rewrite as
[tex]\frac{AB}{BC} = \frac{1}{1}[/tex]
Write as ratio
[tex]AB:BC = 1:1[/tex]
Next is to determine the coordinates of C as follows;
Because B is between both points. we have:
[tex]B(x,y) = (\frac{mx_2 + nx_1}{m+n},\frac{my_2 + ny_1}{m+n})[/tex]
Where
[tex]m:n = AB:BC = 1:1[/tex]
[tex]B(x,y) = B(2,1)[/tex]
[tex]A(x_1,y_1) = A(7,-1)[/tex]
So; we're solving for x2 and y2
[tex]B(2,1) = (\frac{mx_2 + nx_1}{m+n},\frac{my_2 + ny_1}{m+n})[/tex]
Where
Solving for x2;
[tex]x = \frac{mx_2 + nx_1}{m+n}[/tex]
[tex]2 = \frac{1 * x_2 + 1 * 7}{1+1}[/tex]
[tex]2 = \frac{x_2 + 7}{2}[/tex]
Cross Multiply
[tex]2 * 2 = x_2 + 7[/tex]
[tex]4 = x_2 + 7[/tex]
[tex]x_2 = 4 - 7[/tex]
[tex]x_2 = -3[/tex]
Solving for y2;
[tex]y = \frac{my_2 + ny_1}{m+n}[/tex]
[tex]1 = \frac{1 * y_2 + 1 * -1}{1+1}[/tex]
[tex]1 = \frac{y_2- 1}{2}[/tex]
Cross Multiply
[tex]2 * 1 = y_2 - 1[/tex]
[tex]2 = y_2 - 1[/tex]
[tex]y_2 = 2 + 1[/tex]
[tex]y_2 = 3[/tex]
Hence, the coordinates of C are: C(-3,3)