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If point C `(barc)` divides the segment joining the points A(`bara`) and B(`barb`) internally in the ratio m : n, then prove that `barc=(mbarb+nbara)/(m+n)`

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#### Solution

Given that C`(vecc)` divides the segment joining the points A `(veca)` and B `(vecb)`

internally in the ratio m : n

We need to prove that `vecc=(mvecb+nveca)/(m+n)`

Consider the following figure

since `(`

n x length (AC)=m X length (BC)

`nvec(AC)=mvec(CB)`

`n(vec(OC)-vec(OA))=m(vec(OB)-vec(OC))`

`n(vecc-veca)=m(vecb-vecc)`

`nvecc-nveca=mvecb-mvecc`

`(n+m)vecc=mvecb+nveca`

`vecc=(mvecb+nveca)/(m+n)`

Hence proved.

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