+610 Let $F,$ $G,$ and $H$ be collinear points on the Cartesian plane such that $\frac{FG}{GH} = 1.$ If $F = (a, b)$ and $H = (7a, c)$, then what is the x-coordinate of $G$?
+26367 Let $F,$ $G,$ and $H$ be collinear points on the Cartesian plane such that $\frac{FG}{GH} = 1.$
If $F = (a, b)$ and $H = (7a, c)$, then what is the x-coordinate of $G$?
\(\begin{array}{|rcll|} \hline \mathbf{\vec{G}} &\mathbf{=}& \mathbf{(1-\lambda)\vec{F} + \lambda \vec{H}} \\\\ \dfrac{1-\lambda}{\lambda} &=& 1 \\ 1-\lambda &=& \lambda \\ 2\lambda &=& 1 \\ \lambda &=& \dfrac{1}{2}\\\\ \vec{G} &=& \left(1-\dfrac{1}{2}\right)\dbinom{a}{b} + \dfrac{1}{2} \dbinom{7a}{c} \\\\ &=& \dfrac{1}{2} \dbinom{a}{b} + \dfrac{1}{2} \dbinom{7a}{c} \\\\ &=& \dfrac{1}{2} \left( \dbinom{a}{b} + \dbinom{7a}{c} \right) \\\\ &=& \dfrac{1}{2} \dbinom{8a}{b+c} \\\\ &=&\displaystyle \dbinom{4a}{\frac{b+c}{2}} \\\\ \hline \end{array}\)
\(\text{The x-coordinate of $G$ is $4a$}\)
