Line segment division in positive and negative ratios

 

If P is a point dividing segment AB in ratio m:n and it lies between A and B like this:

A____(m)____P____(n)____B

Then position of P will be (m*B + n*A)/(m+n).

For e.g. A is 0 and B is 9. P divides AB into 2:1.

Then P = (1*0 + 2*9)/(2+1) = 18/3 = 6.

A___(6)___P___(3)___B

But if P lies outside AB to the right of B, like this:

A_______(9)___B_____(9)____P

P still divides AB into 2 : 1 since AP/BP = 18/9 = 2/1.

But we can't apply our earlier formula correctly.

So we say that P divides AB into 2: -1 or -2: 1.

Now if we apply the formula:

P = (2*9 + -1*0)/(2-1) = 18
OR
P = (-2*9 + 1*0)/(-2 + 1) = 18

If P lies to the left of A then it can't divide AB into 2: -1 since
AP will always be less than BP.

Now it can divide AB into -1:2 or 1:-2
So
P = (-1*9+2*0)/(2-1) = -9

P____(-9)_____A______(9)____B


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