Question

The coordinates of AB are A(-5, -1) and B(-2, -6). If AC : CB = 3 : 7, what are the coordinates of point C?

I know the answer is (-4.1, -2.5), but I don't understand how to get that answer. If anyone could show me how while explaining the steps, I'd be grateful.

Answer 1

`\text{Let the coordinates of AB are }A(-5,-1), \text{ and }B(-2, -6)\\ \\ \text{let the coordinate of th point C be (x, y), such that}\\ \\ AC:CB=3:7\\ \\ \text{By the section formula, we know that if P(x,y) lies on a line segment AB}\\ \text{where, }A(x_1,y_1), \ B(x_2,y_2)\text{ and satisfy the ratio, }AP:PB=m:n, \text{ then}`

`P=\left ( (mx_2+nx_1)/(m+n), \ (my_2+ny_1)/(m+n) \right ).\\ \\ \text{So using this formula, we get for the given problem}\\ \\ C=\left ( (3(-2)+7(-5))/(3+7), \ (3(-6)+7(-1))/(3+7) \right )\\ \\ \Rightarrow C=\left ( (-6-35)/(10), \ (-18-7)/(10) \right )\\ \\ \Rightarrow C=\left ( (-41)/(10), \ (-25)/(10) \right )\\ \\ \Rightarrow C=\left ( -4.1, \ -2.5 \right )`

Coordinates of the point C is: (-4.1, -2.5)