Midpoint and Distance Between Points

The midpoint and distance formulas are essential coordinate geometry tools. The midpoint gives you the exact centre of a line segment, while the distance formula tells you how far apart two points are. Midpoints appear on both Foundation and Higher papers, while the distance formula (which uses Pythagoras) is more common on Higher. Both are straightforward once you learn the formulas.

What Are the Midpoint and Distance Formulas?

The midpoint of two points is the point exactly halfway between them. You find it by averaging the x-coordinates and averaging the y-coordinates.

The distance between two points is the length of the straight line connecting them. It is found using Pythagoras' theorem applied to the horizontal and vertical differences.

Key Formulas

Step-by-Step Method

Finding the Midpoint

1. Add the two x-coordinates and divide by 2.
2. Add the two y-coordinates and divide by 2.
3. Write the midpoint as a coordinate pair.

Finding the Distance

1. Subtract the x-coordinates to find the horizontal difference.
2. Subtract the y-coordinates to find the vertical difference.
3. Square each difference, add them together, and take the square root.

Worked Example 1 — Foundation Level

Question: Find the midpoint of (2, 6) and (8, 10).

Working:

Midpoint x = (2 + 8)/2 = 10/2 = 5. Midpoint y = (6 + 10)/2 = 16/2 = 8.

Answer: (5, 8)

Worked Example 2 — Higher Level

Question: Find the distance between (1, 3) and (4, 7).

Working:

Horizontal difference: 4 − 1 = 3. Vertical difference: 7 − 3 = 4. Distance = √(3² + 4²) = √(9 + 16) = √25 = 5.

Answer: 5 units

Worked Example 3 — Exam Style

Question: A line segment AB has midpoint M(5, 3). If A = (2, 7), find the coordinates of B.

Working:

Let B = (bx, by). Midpoint x: (2 + bx)/2 = 5, so 2 + bx = 10, giving bx = 8. Midpoint y: (7 + by)/2 = 3, so 7 + by = 6, giving by = −1.

Check midpoint: ((2 + 8)/2, (7 + (−1))/2) = (10/2, 6/2) = (5, 3) ✓

Answer: B = (8, −1)

Common Mistakes

• Subtracting instead of adding for the midpoint. The midpoint formula uses addition of coordinates, not subtraction. Subtraction is used for the distance formula.
• Forgetting to divide by 2. The midpoint averages both coordinates — if you add but do not divide by 2, your answer will be wrong.
• Square root errors in the distance formula. Make sure you square the differences first, add them, and then take the square root of the total — not the square root of each difference separately.

Exam Tips

• The midpoint formula is essentially "find the average of x and the average of y."
• The distance formula is Pythagoras applied to coordinates — if you forget the formula, draw a right-angled triangle on a sketch.
• When given the midpoint and one endpoint, work backwards to find the other endpoint.
• Leave your distance answer as a surd (e.g., √52) unless the question says otherwise, or simplify it (√52 = 2√13).

Practice Questions

Q1 (Foundation): Find the midpoint of (−3, 4) and (5, 2).

Q2 (Foundation): Find the midpoint of (0, −6) and (8, 2).

Q3 (Higher): Find the exact distance between (−1, 2) and (3, −4).

Practise midpoints and distances with instant feedback free on GCSEMathsAI.

Related Topics

• Pythagoras' Theorem
• Linear Graphs and Equation of a Line
• Parallel and Perpendicular Lines

Summary

• The midpoint is found by averaging the x-coordinates and averaging the y-coordinates.
• The distance formula uses Pythagoras: √((x₂ − x₁)² + (y₂ − y₁)²).
• To find a missing endpoint from a midpoint, reverse the formula by doubling and subtracting.
• Midpoints are Foundation and Higher; the distance formula is mainly Higher.
• These formulas connect closely to straight-line graphs and perpendicular bisectors.