Question

Point A is located at ​(4, 8)​ and point B is located at ​(14, 10)​.

What point partitions the directed line segment ​AB into a 1:3 ratio?

a) (6 1/2, 8 1/2)
b) (9, 9)
c) (6, 6)
d) (11 1/2, 9 1/2)

Answer 1

the anwser is a i just took the test and it said that he was wrong

Answer 2

Visualize the problem. We will have to use the distance formula:

d = √(x₂ - x₁)² + (y₂ - y₁)²

Let the unknown point be (x,y). The solution is as follows:

Distance between A and the point.
3 = √(4 - x₁)² + (8 - y₁)²
9 = 16 - 8x₁ + x₁² + 64 - 16y₁ + y₁² --> eqn 1

Distance between point and B
1 = √(14 - x₁)² + (10 - y₁)²
1 = 196 - 28x₁ + x₁² + 100 - 20y₁ + y₁² --> eqn 2

Subtract eqn 2 from eqn 1:
8 = -180 + 20x₁ - 36 + 4y₁
8 = -216 + 20x₁ + 4y₁
224 = 20x₁ + 4y₁
56 = 5x₁ + y₁ --> eqn 3

The last equation would be the linear equation using points A and B.
m = (10-8)/(14-4) = 1/5 = (8 - y₁)/(4 - x₁)
4 - x₁ = 5(8 - y₁)
4 - x₁ = 40 - 5y₁
-36 = x₁ - 5y₁ --> eqn 4

Solve equations 3 and 4 simultaneously:
x₁ = 9.38
y₁ = 9.08
Therefore, the closest answer is letter B.