Question

find the coordinates of the point on the directed segment from (3,2) to (6,8) that divides it into a ratio of 1:3.​

Answer 1

`\fbox{\begin{minipage}{9em}Option d is correct\end{minipage}}`

Explanation:

Given:

Point A(3, 2)

Point B(6, 8)

Point C lies on segment line AB and divides AB into a ratio of 1 : 3

Solve for:

Coordinate of C

Solution:

Denote the coordinate of origin O(0, 0)

Point A(3, 2) => Vector OA(3 - 0, 2 - 0) or OA(3, 2)

Point B(6, 8) => Vector OB(6 - 0, 8 - 0) or OB(6, 8)

=> Vector AB = Vector OB - Vector OA = (6 - 3, 8 - 2) or vector AB(3, 6)

Point C lies on segment line AB and divides AB into a ratio of 1 : 3

=> Vector AC = (1/3) Vector CB, or

AC/CB = 1/3

=> AC/(AC + CB) = 1/(1 + 3)

=> AC/AB = 1/4

=> AC = AB/4

=> Vector AC = (1/4) Vector AB = ((1/4)*3, (1/4)*6) = (0.75, 1.5)

=> Vector OC = Vector OA + Vector AC = (3 + 0.75, 2 + 1.5) = (3.75, 3.5)

=> C(3.75, 3.5)

Option d is correct!

Answer 2

(3.75, 3.5)

Explanation:

The weighting factors for the weighted average of the end points are the reverse of the length ratios.

For a 1 : 3 split, the first point is weighted by a factor of 3; the second point is weighted by a factor of 1. The weighted sum is divided by the total of the weights.

(3(3, 2) +1(6, 8))/(3+1) = (9+6, 6+8)/4 = (3.75, 3.5)