Question

A, b, and c are collinear, and B is between a and c. The ratio of AB to AC is 1:2. If A is at (7,-1) and B is at (2,1) what are the coordinates of point C

Answer 1

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;

$AB:AC = 1:2$

Convert to division

$\frac{AB}{AC} = \frac{1}{2}$

AC = AB + BC;

$\frac{AB}{AB + BC} = \frac{1}{2}$

Cross Multiply

$2 * AB = 1 * (AB + BC)$

$2 AB = AB + BC$

$2AB - AB = BC$

$AB = BC$

Divide both sides by BC

$\frac{AB}{BC} = 1$

Rewrite as

$\frac{AB}{BC} = \frac{1}{1}$

Write as ratio

$AB:BC = 1:1$

Next is to determine the coordinates of C as follows;

Because B is between both points. we have:

$B(x,y) = (\frac{mx_2 + nx_1}{m+n},\frac{my_2 + ny_1}{m+n})$

Where

$m:n = AB:BC = 1:1$

$B(x,y) = B(2,1)$

$A(x_1,y_1) = A(7,-1)$

So; we're solving for x2 and y2

$B(2,1) = (\frac{mx_2 + nx_1}{m+n},\frac{my_2 + ny_1}{m+n})$

Where

Solving for x2;

$x = \frac{mx_2 + nx_1}{m+n}$

$2 = \frac{1 * x_2 + 1 * 7}{1+1}$

$2 = \frac{x_2 + 7}{2}$

Cross Multiply

$2 * 2 = x_2 + 7$

$4 = x_2 + 7$

$x_2 = 4 - 7$

$x_2 = -3$

Solving for y2;

$y = \frac{my_2 + ny_1}{m+n}$

$1 = \frac{1 * y_2 + 1 * -1}{1+1}$

$1 = \frac{y_2- 1}{2}$

Cross Multiply

$2 * 1 = y_2 - 1$

$2 = y_2 - 1$

$y_2 = 2 + 1$

$y_2 = 3$

Hence, the coordinates of C are: C(-3,3)