Question

Item 11 Which points divide overline AB into segments with lengths in the ratios 1:3 or 3:1?

Answer 1

(9, 2)

Explanation:

What point divides the directed line segment ​ AB¯¯¯¯¯ ​ ⁢ into a 3:1 ratio?

coordinate plane with segment A B with A at (0, 2) and B at (12, 2)

(3, 2)

(4, 2)

(8, 2)

(9, 2)

Solution:

If point O(x, y) divides the line segment AB with endpoints A(
`x_1,y_1`) and B(
`x_2,y_2`) in the ratio n:m, the coordinates of O is:

`x=(n)/(n+m) (x_2-x_1)+x_1\\\\y=(n)/(n+m) (y_2-y_1)+y_1`

Let us assume C(x, y) is the point that divides segment A B with A at (0, 2) and B at (12, 2) in the ratio 3 : 1. Hence:

`x=(3)/(3+1)(12-0)+0=(3)/(4)(12)=9 \\\\y=(3)/(3+1)(2-2)+2=(3)/(4)(0)+2=2`

Therefore the coordinate of point C is at (9, 2).