Question

Point B is 2/3 the way from A(-2,5)to C(4,-4). Find the location of B

Answer 1

well, we'll first off put the point AC in component form by simply doing a subtraction of C - A, multiply that by the fraction 2/3, and that result will get added to point A, to get point B.

$\bf \textit{internal division of a segment using a fraction}\\\\ A(\stackrel{x_1}{-2}~,~\stackrel{y_1}{5})\qquad C(\stackrel{x_2}{4}~,~\stackrel{y_2}{-4})~\hfill \frac{2}{3}\textit{ of the way from }A\to C \\\\[-0.35em] ~\dotfill\\\\ (\stackrel{x_2}{4}-\stackrel{x_1}{(-2)}, \stackrel{y_2}{-4}-\stackrel{y_1}{5})\implies (4+2,-9) \stackrel{\textit{component form of segment AC}}{\qquad \implies \qquad (6,-9)} \\\\[-0.35em] ~\dotfill$

$\bf \stackrel{~\hfill \textit{coordinates to be added to point A}} {\cfrac{2}{3}(6,-9)\implies \cfrac{2}{3}(6)~,~\cfrac{2}{3}(-9)\qquad \implies \qquad \left(4,-6\right)} \\\\\\ \stackrel{\textit{additions to point A}} {\stackrel{\textit{point A}}{(-2,5)}+\left( 4,-6\right)}\implies \left( -2+4~~,~~5-6\right) = B\implies (2~,~-1)=B$