$a\in\mathbb{Z}/\mathbb{Z}_{p}$ and Triangles

Question

Can someone please give me a geometric explanation for the following: if you consider the integers modulo a prime as a group under + and plot the set of points $(a,a')$ where $a\in\mathbb{Z}/\mathbb{Z}_{p}$ and a' is its additive inverse, the graph you get is a triangle? -Thanks

Answer 1

Essentially, all the points you get other than the origin $(0,0)$ are colinear as $a^{-1}=p-a$ for $a\in\{1,\ldots,p-1\}$. So they form a line with gradient -1 (that is, the line given by $y=p-x$) and you get the triangle given by $(0,0)$, $(1,p-1)$ and $(p-1,1)$.