Core answer: for points (x₁,y₁) and (x₂,y₂), slope k=(y₂−y₁)/(x₂−x₁) and distance d=√[(x₂−x₁)²+(y₂−y₁)²]; two lines are perpendicular exactly when their slopes multiply to −1; the proportion a:b=c:d holds exactly when cross-products match (ad=bc).
Three formulas at a glance
| Quantity | Formula | Reading |
|---|---|---|
| Slope | k=(y₂−y₁)/(x₂−x₁) | rise per unit of run |
| Distance | d=√(Δx²+Δy²) | Pythagoras in coordinates |
| Midpoint | ((x₁+x₂)/2, (y₁+y₂)/2) | average each component |
Edge cases: x₁=x₂ gives a zero denominator — the line is vertical and slope is undefined; y₁=y₂ gives k=0 — horizontal.
Slope-to-grade conversion table
Engineering “grade” = slope × 100% (or 1:n):
| Scenario | Slope | Grade | Approx angle |
|---|---|---|---|
| Wheelchair ramp limit | 0.083 | 8.3% (1:12) | 4.8° |
| Parking garage ramp | 0.15 | 15% | 8.5° |
| Typical staircase | 0.6–0.8 | 60–80% | 31–39° |
| Highway max grade | 0.03–0.06 | 3–6% | 1.7–3.4° |
| 45° diagonal | 1 | 100% | 45° |
Core ratio properties
- Cross-multiply: a:b = c:d ⇔ ad = bc — the only rule you need for checking and solving
- Simplify: divide every term by the GCD — 12:18 = 2:3
- Chained ratios: a:b = 2:3 and b:c = 4:5 give a:b:c = 8:12:15 (align b to the LCM 12)
- Proportional split: total × each share. ¥90 split 2:3:4 → 20, 30, 40
Example: auditing a wheelchair ramp
Code caps wheelchair ramps at 1:12. On site: 4.8m of horizontal run, 0.45m of rise.
k = 0.45÷4.8 = 0.0937 = 9.4% > 8.3% (1:12) — fails; the run must extend to at least 0.45×12=5.4m. The actual ramp surface length comes from the distance formula: √(5.4²+0.45²) ≈ 5.42m — order materials to that.
Example: map distance and scale
A 1:50,000 map shows 6.4cm between two points:
Ground distance = 6.4 × 50,000 = 320,000cm = 3.2km. Given grid coordinates instead, say (1200, 800) and (1800, 1500) meters: d=√(600²+700²)=√850,000≈922m — coordinate math and scale math complement each other.
Common mistakes
- Mismatched subtraction order: (y₂−y₁)/(x₂−x₁) and (y₁−y₂)/(x₁−x₂) are both right; mixing them flips the sign.
- “A 10% grade is 10°”: 10% means tanθ=0.10, so θ≈5.7°. Percent and degrees only track at small grades (100%=45°).
- “a:b=c:d means a=c”: equality is of ratios, not terms — 2:3=4:6 yet 2≠4.
- Forgetting the square root: √(Δx²+Δy²) without the root gives distance squared — 922²=850,000m is obviously absurd; sanity-check units.