Does it matter which two points you use to find a slope? No, and scale drawings explain why.
Por qué funciona
Draw two slope triangles on the same line. Their bottom sides are both horizontal, their upright sides are both vertical, and their long sides lie along the same line. So their angles match, and one triangle is a scaled copy of the other, like a scale drawing. Scaling multiplies every side by the same factor $k$, so $$\frac{\text{rise}_2}{\text{run}_2} = \frac{k \cdot \text{rise}_1}{k \cdot \text{run}_1} = \frac{\text{rise}_1}{\text{run}_1}$$
Idea clave
All slope triangles on one line are similar, so they all give the same rise ÷ run. That is why a line has a single slope, and why any two of its points give it.
Worked examples
Ejemplo resuelto: Scale a triangle
On a line, one slope triangle has rise 2 and run 3. A bigger slope triangle on the same line has run 12. What is its rise?
- The run grew from 3 to 12, a scale factor of 4.
- The rise grows by the same factor: $2 \cdot 4 = 8$.
- Check: $\frac{8}{12} = \frac{2}{3}$.
Respuesta: 8
Ejemplo resuelto: Same line or not?
Could a triangle with rise 4 and run 6 and another with rise 6 and run 9 be slope triangles on the same line?
- $\frac{4}{6} = \frac{2}{3}$ and $\frac{6}{9} = \frac{2}{3}$.
- The ratios match, so yes (if the lines also share a point).
Respuesta: yes
Error común
Thinking a bigger slope triangle means a steeper line. A bigger triangle on the same line has a bigger rise and a bigger run in the same proportion, so the slope is unchanged.