Question

Segment YZ has endpoints (6, 3) and (3, 9). Find the coordinates of the point that divides the line segment directed from Y to Z in the ratio of 2:1.

Answer 1

`(4,7)`

Explanation:

We are asked to find the coordinates of the the point that divides the line segment directed from Y to Z in the ratio of 2:1.

`Y(6,3)=(x_1,y_1)` and
`Z(3,9)=(x_2,y_2)`

We will use section formula to solve our given problem.

When a point P divides a line segment AB internally in the ratio m:n, then

`[x=(mx_2+nx_1)/(m+n),y=(my_2+ny_1)/(m+n)]`

`[x=(2\cdot 3+1\cdot 6)/(2+1),y=(2\cdot 9+1\cdot 3)/(2+1)]`

`[x=(6+6)/(3),y=(18+3)/(3)]`

`[x=(12)/(3),y=(21)/(3)]`

`[x=4,y=7]`

Therefore, the coordinates of point would be
`(4,7)`.

Answer 2

(4, 7)

Explanation:

The point of interest is ...

P = (2Z +1Y)/(2+1) = ((2·3+6)/3, (2·9+3)/3)

P = (4, 7)

__

The point that divides the segment into the ratio a:b is the weighted average of the endpoints, with the weights being "b" and "a". The weight of the first end point corresponds to the length of the far end of the segment.