# Sum of x-coordinates

Regular hexagon HEXAGN has side length 5 and center at the origin. Sides HE and AG are parallel to the x-axes. Transltion T maps HEXAGN to H'E'X'A'G'N' such that N' and H are the same point. What is the sum of the x-coordinates of points H',E',X',A',G', and N'?

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The original hexagon, HEXAGN, has $x$-coordinates that sum to zero by reflection symmetry across the $y$-axis. The uniform translation means that all six points have their $x$-coordinates increased by the same value. What is that value? $N$ was the leftmost point and $H$ was the bottom-left point. What is the horizontal distance between them?
Draw the right triangle with hypotenuse $NH$, length $5$. The short leg of that triangle has length $5/2$, so the answer is $6\cdot 5/2=15$. In slightly more detail, $$\sum x'=\sum x+\sum(x'-x)=0+\sum\frac{5}{2}=15.$$