Math

Distance and Midpoint Formulas

2 min readEasy5-question drill

Two points on a grid, and the test wants to know how far apart they are or what's exactly between them. These two formulas turn that into quick arithmetic — and they show up in coordinate geometry and circle problems too.

Where this lesson fits

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→Coordinate Geometry →Circles →Right Triangles and Trigonometry

A coordinate plane is just a grid with a horizontal x-axis and a vertical y-axis. Every point is written as (x, y) — the first number tells you how far right or left, the second how far up or down. Two formulas let you measure relationships between points.

The Distance Formula finds the straight-line distance between two points. If your points are (x₁, y₁) and (x₂, y₂):

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

The little subscripts (the 1 and 2) are just labels so you can tell the two points apart — point 1 and point 2. This formula is secretly the Pythagorean theorem. The horizontal gap between the points is one leg of a right triangle, the vertical gap is the other leg, and the distance is the hypotenuse.

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Distance is the hypotenuse of a right triangle whose legs are the horizontal and vertical gaps.

So to use it:

Subtract the x-values, square the result.
Subtract the y-values, square the result.
Add those two squares, take the square root.

Example: distance from (0,0) to (3,4) is √[(3−0)² + (4−0)²] = √[9+16] = √25 = 5.

The Midpoint Formula finds the point exactly halfway between two points. You just average the x-values and average the y-values:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Example: midpoint of (2, 4) and (6, 10) is ((2+6)/2, (4+10)/2) = (4, 7).

Here's the move the test loves: they give you the midpoint and one endpoint, then ask for the other endpoint. Don't panic — just set up the formula and solve. If the midpoint's x is 3 and one endpoint's x is 1, then 3 = (1 + x)/2. Multiply both sides by 2: 6 = 1 + x, so x = 5. Do the same for y.

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The midpoint M sits exactly halfway between endpoints A and B.

A quick mental check: distance uses subtraction and a square root; midpoint uses addition and division by 2. Mixing them up is the #1 error, so anchor on that.

Quick check

What is the distance between (0, 0) and (3, 4)?

Worked examples

Example 1

What is the distance between the points (1, 2) and (4, 6)?

Example 2

The midpoint of segment AB is M(3, 7). If A = (1, 4), what are the coordinates of B?

Example 3

The midpoint of segment PQ is M(5, −2). If P = (−1, 6), what is the sum of the coordinates of Q? (Enter your answer.)

Common pitfalls

Forgetting the square root in distance

After adding the squares you get a number like 25, and it's tempting to stop. But distance is √25 = 5. The unsquared sum is almost always a wrong-answer choice.

Mixing up the two formulas

Distance subtracts, squares, adds, and roots. Midpoint adds and divides by 2. Anchor on 'midpoint = average' so you never divide in a distance problem.

Sign errors with negative coordinates

When a coordinate is negative, like (−1, 6), subtraction becomes addition: 4 − (−1) = 5. Write the substitution out fully instead of doing it in your head.

Solving for the midpoint when they gave it to you

When the problem gives the midpoint and one endpoint, you're solving for the other endpoint — set up midpoint = (known + unknown)/2 and solve, don't just average the two points you see.

Key takeaways

Distance: d = √[(x₂−x₁)² + (y₂−y₁)²] — it's the Pythagorean theorem in disguise.
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2) — just average each coordinate.
To find a missing endpoint, set up midpoint = (known + unknown)/2 and solve each coordinate separately.
Always finish distance with the square root; the unsquared sum is a trap answer.
Substitute negative coordinates carefully — subtracting a negative becomes adding.

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Try it yourself

5 practice questions on Distance and Midpoint Formulas, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Distance and Midpoint Formulas

You are here

→Coordinate Geometry →Circles →Right Triangles and Trigonometry