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Daily Math Minute

Coordinate Geometry

Distance & Midpoint Formulas

Using the distance and midpoint formulas on the coordinate plane.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Turning Geometry into Algebra

You already found coordinate distances in Grade 8 using the Pythagorean Theorem directly. Before reading on, think about the exact midpoint between two coordinate points — predict a formula for it, and think about why simply averaging the two x-coordinates (and separately averaging the two y-coordinates) should locate that midpoint exactly.

Definition — Distance and Midpoint Formulas

For two points (x₁, y₁) and (x₂, y₂): the distance between them is d = √((x₂ − x₁)² + (y₂ − y₁)²), and their midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

The distance formula is exactly the Pythagorean Theorem, generalized into a reusable formula: the horizontal and vertical coordinate differences form the two legs of a right triangle, and the direct distance between the points is that triangle's hypotenuse.

Averaging works for the midpoint because a midpoint must be exactly halfway along each direction independently — halfway between the two x-coordinates, and, separately, halfway between the two y-coordinates. The average of two numbers is precisely the value exactly halfway between them, so averaging each coordinate separately locates the point that's simultaneously halfway across and halfway up (or down) — exactly the midpoint.

Worked Example — Finding the Distance Between Two Points

Find the distance between (2, 3) and (7, 15). d = √((7 − 2)² + (15 − 3)²) = √(25 + 144) = √169 = 13.

Worked Example — Finding a Midpoint

Find the midpoint of the segment connecting (−4, 6) and (10, 2). Midpoint = ((−4 + 10)/2, (6 + 2)/2) = (6/2, 8/2) = (3, 4).

Worked Example — Verifying a Point Is Equidistant

Is the point (1, 1) equidistant from A(−2, 5) and B(4, 5)? Distance to A: √((1−(−2))² + (1−5)²) = √(9+16) = 5. Distance to B: √((1−4)² + (1−5)²) = √(9+16) = 5. Both distances are 5, so (1, 1) is equidistant from A and B — meaning it lies on segment AB's perpendicular bisector.
d=(x2−x1)2+(y2−y1)2M=(x1+x22,y1+y22)d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \qquad M = \left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)

Graph Visualizer

Domain & range
2
Evaluate a point
  • x^2 = 0

Tip

The order you subtract coordinates in doesn't matter for distance, since the difference gets squared either way — but it does matter for slope, which doesn't square its differences, so keep that distinction in mind when switching between these related coordinate-plane formulas.

Common Mistakes

  • Forgetting to take the square root at the end of the distance formula, leaving the squared distance as a final answer.

    The formula under the radical gives the squared distance — always take the final square root to report the actual distance itself.

  • Averaging only the x-coordinates or only the y-coordinates when finding a midpoint, instead of both.

    A midpoint's coordinates come from averaging the x-values together and separately averaging the y-values — both averages are needed to fully locate the point.

Key Takeaways

  • The distance formula is the Pythagorean Theorem applied directly to coordinate differences.
  • The midpoint formula averages each coordinate separately, since a midpoint sits exactly halfway in each direction independently.
  • These two formulas connect coordinate algebra to the same geometric ideas — distance and equidistance — used throughout this course.

Summary

Distance and midpoint formulas turn geometric facts into calculations. The final lesson puts them to work proving general geometric theorems using coordinates and algebra.

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