Question

Given A(2,12) and B(5,0), find the coordinates of P such that P separates segment AB into two parts with lengths in a ratio of 2 to 1?

Answer 1

The target point is `(x,y)=(4,4)`

Explanation:

Assumption: the two parts is to the left side of the line

Let point 1 be `P_1->(x_1,y_1)=(2,12)`
Let point 2 be `P_2->(x_2,y_2)=(5,0)`
Let the target point be `P_t->(x_t,y_t)`

Combining the 2 parts and 1 part give a total of 3 parts. So as a fraction of the whole the 2 parts is `2/3`

So

`x_t" "=" "x_1+2/3(x_2-x_1)" " =" " 2+2/3(5-2)" " =" " 4`
`y_t" "=" "y_1+2/3(y_2-y_1)" "=" "12+2/3(0-12)" "=" "4`

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Check:

`P_1->P_t = sqrt(2^2+8^2)`

`P_t->P_2=sqrt(4^2_1^2)`

So `(sqrt(2^2+8^2))/(sqrt(2^2+8^2)+ sqrt(4^2+1^2)) = 0.66bar6=2/3`