Lab: Ratio Explorer
Slide the unit x and watch every side of both special triangles relabel itself — always in exact radical form.
45-45-90 triangle
The triangle keeps its shape — only the labels change with x.
| Side | Exact | ≈ Decimal |
|---|
Controls
The unit x scales the whole ratio.
Experiment 1 — spot the pattern. Move x from 1 to 12 and watch one column of the table.
Every entry is a fixed multiple of x: the ratio never changes.
Experiment 2 — Pythagoras check. With x = 5 in the 30-60-90 tab: 5² + (5√3)² = 25 + 75 = 100 = 10².
The ratio satisfies the Pythagorean theorem exactly — that is where it comes from.