Division of an interval in a given ratio. Suppose given an interval М1М2. Let us find the coordinates a of point М on the interval for which .

Compose the vectors and .

y

M₂(x₂,y₂)

M(x,y)

M₁(x₁,y₁)

0 x

It is known from vector algebra that the condition for two vectors to be collinear is the proportionality of their respective coordinates:

.

From the first fraction we obtain

; ; .

This gives the x coordinate; y is found in a similar way:

; .

To obtain a formula for the midpoint of the interval, we take l=1:

; .

Example. Given the two points М₁(–2;4) and М₂(6;2), find the midpoint of the interval М₁, М₂.

М₁ ,

М₂ .