Two points on a grid. How far apart are they? What’s exactly halfway between them? These two questions — answered by the distance formula and the midpoint formula — show up everywhere: GPS navigation, video game graphics, construction layouts, and half of coordinate geometry exams. Both formulas are simpler than they look, and both trace back to one idea you already know.
The Distance Formula
The distance between points (x₁, y₁) and (x₂, y₂) is:
d = √((x₂ – x₁)^2 + (y₂ – y₁)^2)
This is the Pythagorean theorem in disguise. The horizontal gap (x₂ − x₁) and the vertical gap (y₂ − y₁) form the legs of a right triangle, and the segment joining the points is the hypotenuse. You’re just computing that hypotenuse.
Worked Example 1: Basic distance
Find the distance between A(2, 3) and B(6, 6).
- Label the coordinates: x₁ = 2, y₁ = 3, x₂ = 6, y₂ = 6
- Find the differences: x₂ − x₁ = 6 − 2 = 4; y₂ − y₁ = 6 − 3 = 3
- Square each: 4² = 16; 3² = 9
- Add and root: d = √(16 + 9) = √25 = 5 units
Worked Example 2: Distance with negative coordinates
Find the distance between C(−1, 4) and D(5, −4).
- Differences: x₂ − x₁ = 5 − (−1) = 6; y₂ − y₁ = −4 − 4 = −8
- Square each: 6² = 36; (−8)² = 64
- Add and root: d = √(36 + 64) = √100 = 10 units
Notice the negative difference became positive the moment it was squared — distance is never negative, so that’s exactly what should happen.
The Midpoint Formula
The midpoint is the point exactly halfway along a segment. Instead of measuring anything, just average the x’s and average the y’s:
M = ((x₁ + x₂)/(2), (y₁ + y₂)/(2))
Distance gives you a single number (a length). Midpoint gives you an ordered pair (a location). If your “midpoint” came out as one number, you’ve mixed up the two formulas.
Worked Example 3: Finding the midpoint
Find the midpoint of the segment joining P(−4, 2) and Q(6, 8).
- Average the x-values: (−4 + 6) ÷ 2 = 2 ÷ 2 = 1
- Average the y-values: (2 + 8) ÷ 2 = 10 ÷ 2 = 5
- Combine: midpoint = (1, 5)
Worked Example 4: Solving for a missing endpoint
A classic exam twist: you’re given the midpoint and one endpoint, and must find the other. Segment AB has midpoint M(3, 4) and endpoint A(1, 2). Find B.
- Set up the x-equation: (1 + x₂) ÷ 2 = 3 → multiply by 2: 1 + x₂ = 6 → x₂ = 5
- Set up the y-equation: (2 + y₂) ÷ 2 = 4 → multiply by 2: 2 + y₂ = 8 → y₂ = 6
The missing endpoint is B(5, 6). Quick check: the average of 1 and 5 is 3, and of 2 and 6 is 4 — it lands exactly on M. Always verify this way.
3 Mistakes to Avoid
- Mixing up the signs: distance subtracts inside the parentheses (x₂ − x₁); midpoint adds (x₁ + x₂). Swapping them is the #1 error on this topic.
- Leaving a negative under the square root: (y₂ − y₁)² is always non-negative. If you have a negative under the radical, you forgot to square first.
- Answering a midpoint question with a length: the midpoint is a point (x, y), not a number. Write both coordinates.
Pro move for word problems: sketch the points first. A rough plot catches labeling slips (swapped x and y) before they cost you the whole problem.