Question

Point C open parentheses 4 comma space 2 close parentheses divides the line segment joining points A open parentheses 2 comma space minus 1 close parentheses and B open parentheses x comma space y close parentheses such that A C : C B equal 3 : 1. What are the coordinates of point B?

Answer 1

Question:

Point C (4,2) divides the line segment joining points A(2,-1) and B(x, y) such that AC:  CB = 3:1.

What are the coordinates of point B?

Answer:

$B = (\frac{14}{3},3)$

Step-by-step explanation:

Given

$C = (4,2)$

$A = (2,-1)$

$AC : CB = 3 : 1$

Required

Find the coordinates of B

Coordinates of a line segment is calculated using:

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

In this case:

$(x,y) = (4,2)$

$AC:CB = m : n = 3:1$

$A = (2,-1)$ --- $(x_1,y_1)$

The equation becomes

$(4,2) = (\frac{3x_2+2}{1+3}, \frac{3y_2 - 1}{1+3})$

So, the coordinates of B is: $(x_2,y_2)$

Solving further

$(4,2) = (\frac{3x_2+2}{4}, \frac{3y_2 - 1}{4})$

Multiply through by 4

$4 * (4,2) = (\frac{3x_2+2}{4}, \frac{3y_2 - 1}{4}) * 4$

$(16,8) = (3x_2+2, 3y_2 - 1)$

By comparison:

$3x_2 + 2 = 16$

$3y_2 - 1 = 8$

So:

$3x_2 + 2 = 16$

$3x_2 = 16 - 2$

$3x_2 = 14$

$x_2 = \frac{14}{3}$

$3y_2 - 1 = 8$

$3y_2 = 8+1$

$3y_2 = 9$

$y_2 = 3$

The coordinates of B is:

$B = (\frac{14}{3},3)$