Question

Find the coordinates of point, P, which divides the line segment from A = (4, 0) to B = (6, 8) in a ratio of 1:2.

Answer 1

Then point P is (
`(14)/(3), (8)/(3)`)

The correct answer is option 2.

Explanation:

First let's calculate the distance between points A and B

A: (4.0)

B: (6.8)

The distance in x between both is:

`X_2-X_1 = 6-8 = 2`

The distance in the y axis between both is:

`y_2-y_1 = 8-0 = 8`.

We know that the point divides the line segment in a 1: 2 ratio as shown in the following diagram

A --------- P ------------------- B

Where the distance PB is twice as large as the AP distance.

Then
`AP = (1)/(3)AB\\\\PB = (2)/(3)AB.`

Therefore Point P is at a distance d = 1/3 * (
`X_(2)-X_(1)`) from point A.

So:

`P_(x) = 4 + (1)/(3)2 = (14)/(3)\\\\P_(y) = 0 + (1)/(3)8 = (8)/(3)`

Then point P is (
`(14)/(3), (8)/(3)`)

The correct answer is option 2.

Answer 2

Given the points

A(x₁,y₁) = (4,0)

B(x₂,y₂) = (6,8)

ratio l:m = 1:2

The formula that we use to find the point P that divides the given line segment in the ratio l:m is

P(x,y) =
`((lx2+mx1)/(l+m) ,(ly2+my1)/(l+m) )`

=
`((1.6+2.4)/(1+2) , (1.8+2.0)/(1+2) )`

=
`((6+8)/(3) ,(8+0)/(3) )`

=
`((14)/(3), (8)/(3) )`

Hence the right option is B)