Question

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 $G$?

Answer 1

F, G, and H all lie on the same line. If FG/GH = 1, then FG = GH, which is to say the distance between F and G is equal to the distance between G and H. This means either F and H are the same point, or G is the midpoint of F and H.

They're not the same point, because the x-coordinate of H is 7 times that of F. So G must be halfway between F and H.

Then the x-coordinate of G is

(a + 7a)/2 = 8a/2 = 4a