Question

What are the coordinates of the point on the directed line segment from

(
−
8
,
8
)
(−8,8) to
(
−
2
,
−
10
)
(−2,−10) that partitions the segment into a ratio of 1 to 2?

Answer 1

so let's say A(-3 , -10) and B(9 , 5), so that point C partitions it in a 1 : 2 ratio from A to B

`\textit{internal division of a line segment using ratios} \\\\\\ A(-3,-10)\qquad B(9,5)\qquad \qquad \stackrel{\textit{ratio from A to B}}{1:2} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{1}{2}\implies \cfrac{A}{B} = \cfrac{1}{2}\implies 2A=1B\implies 2(-3,-10)=1(9,5)`

`(\stackrel{x}{-6}~~,~~ \stackrel{y}{-20})=(\stackrel{x}{9}~~,~~ \stackrel{y}{5}) \implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-6 +9}}{1+2}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{-20 +5}}{1+2} \right)} \\\\\\ C=\left( \cfrac{ 3 }{ 3 }~~,~~\cfrac{ -15}{ 3 } \right)\implies C=(1~~,~-5)`