Slope of y = ½x using two different slope triangles
On a straight line, the slope is the same no matter which two points you pick. Let's see why, using similar triangles.
Here are our axes — a coordinate grid.
And here's a straight line passing through these points.
Pick two close points. Drop straight down, then across — that makes a slope triangle. Rise is one, run is two.
So slope here equals rise over run: one over two.
Now pick two points farther apart. A new slope triangle: rise is two, run is four.
Both triangles have the same three angles — they are similar triangles!
The slope from this triangle is two over four.
And two over four simplifies to one over two — exactly the same as before.
Similar triangles prove rise over run never changes. That's why slope is one number for the whole line.