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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 $G$?

gueesstt  Apr 13, 2018
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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 $G$?

$$\begin{array}{|rcll|} \hline \mathbf{\vec{G}} &\mathbf{=}& \mathbf{(1-\lambda)\vec{F} + \lambda \vec{H}} \\\\ \dfrac{1-\lambda}{\lambda} &=& 1 \\ 1-\lambda &=& \lambda \\ 2\lambda &=& 1 \\ \lambda &=& \dfrac{1}{2}\\\\ \vec{G} &=& \left(1-\dfrac{1}{2}\right)\dbinom{a}{b} + \dfrac{1}{2} \dbinom{7a}{c} \\\\ &=& \dfrac{1}{2} \dbinom{a}{b} + \dfrac{1}{2} \dbinom{7a}{c} \\\\ &=& \dfrac{1}{2} \left( \dbinom{a}{b} + \dbinom{7a}{c} \right) \\\\ &=& \dfrac{1}{2} \dbinom{8a}{b+c} \\\\ &=&\displaystyle \dbinom{4a}{\frac{b+c}{2}} \\\\ \hline \end{array}$$

$$\text{The x-coordinate of G is 4a}$$

heureka  Apr 13, 2018