Question

25. Determine the points of a point P that is two times closer to B(15. 12)

as it is to A(6, 0).

Answer 1

Check the picture below.

so we can say that point P cuts the segment AB to a ratio of 2:1 from A to B, thus

`\textit{internal division of a line segment using ratios} \\\\\\ A(6,0)\qquad B(15,12)\qquad \qquad \stackrel{\textit{ratio from A to B}}{2:1} \\\\\\ \cfrac{A\underline{P}}{\underline{P} B} = \cfrac{2}{1}\implies \cfrac{A}{B} = \cfrac{2}{1}\implies 1A=2B\implies 1(6,0)=2(15,12)`

`(\stackrel{x}{6}~~,~~ \stackrel{y}{0})=(\stackrel{x}{30}~~,~~ \stackrel{y}{24}) \implies P=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{6 +30}}{2+1}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{0 +24}}{2+1} \right)} \\\\\\ P=\left( \cfrac{ 36 }{ 3 }~~,~~\cfrac{ 24}{ 3 } \right)\implies P=(12~~,~~8)`