04 // Interactive Formula Compendium // GEOMETRY-TRIGd = √[ (x₂ − x₁)² + (y₂ − y₁)² ]
Distance, Midpoint & Line Slope Formula — Coordinate Geometry Sheet
In a 2D Cartesian coordinate plane, the Euclidean distance between P₁(x₁, y₁) and P₂(x₂, y₂) is d = √[(x₂ − x₁)² + (y₂ − y₁)²], and their equidistant midpoint is M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
Formula Reference
Main Formula
d = √[ (x₂ − x₁)² + (y₂ − y₁)² ]
Rearranged Forms
M = ( (x₁ + x₂) / 2, (y₁ + y₂) / 2 ) (Midpoint Coordinates)
m = (y₂ − y₁) / (x₂ − x₁) (Slope / Gradient)
P(t) = ( (1 − t)x₁ + t x₂, (1 − t)y₁ + t y₂ ) (Section Formula)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| d | Euclidean Distance (d) | units |
| x1 | Point 1 X-Coordinate (x₁) | 1 |
| y1 | Point 1 Y-Coordinate (y₁) | 1 |
| x2 | Point 2 X-Coordinate (x₂) | 1 |
| y2 | Point 2 Y-Coordinate (y₂) | 1 |
Inputs & Parameters
[1]
[1]
[1]
[1]
Calculated Result (Euclidean Distance (d))
d =10units
Formula Notes & Background
- Construct a right triangle with horizontal leg Δx = |x₂ − x₁| and vertical leg Δy = |y₂ − y₁|.
- Apply the Pythagorean theorem d² = (Δx)² + (Δy)² to the hypotenuse connecting (x₁, y₁) and (x₂, y₂).
- For the midpoint, congruent similar triangles of half-height and half-width place M at the arithmetic mean of each coordinate.
Step-by-Step Examples
#1: Find the distance, midpoint, and slope between P₁(1, 2) and P₂(7, 10).
(x₁, y₁) = (1, 2)(x₂, y₂) = (7, 10)
d = √[(7 − 1)² + (10 − 2)²] = √(36 + 64) = √100
→ d = 10, Midpoint M = (4, 6), Slope m = 8/6 = 4/3
#2: Find the perpendicular bisector of the segment from (−2, 3) to (4, −1).
M = ((−2+4)/2, (3−1)/2) = (1, 1)m = (−1−3)/(4−(−2)) = −2/3
Perpendicular slope m_⊥ = −1/m = 3/2; line through (1, 1): y − 1 = (3/2)(x − 1)
→ y = 1.5x − 0.5
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Frequently Asked Questions
How does the Euclidean distance formula extend to 3D and n-dimensional vectors?+
In ℝⁿ, the L₂ Euclidean distance between vectors u and v is d(u, v) = √[Σ_{i=1..n} (v_i − u_i)²].
What is the difference between Euclidean (L₂) and Manhattan (L₁) distance?+
Euclidean distance measures the straight-line crow-flies hypotenuse √(Δx² + Δy²), while Manhattan taxicab distance sums axis-aligned grid steps |Δx| + |Δy|.
Does the order of points (P₁ vs P₂) affect the distance or midpoint?+
No, because squaring (x₂ − x₁)² = (x₁ − x₂)² and averaging (x₁ + x₂)/2 are both symmetric operations.