Worked example 1
Find a distance
Find the distance between A(-2, 1) and B(4, 9).
- The coordinate changes are 4 − (-2) = 6 and 9 − 1 = 8.
- Use distance = √(6² + 8²) = √100.
Answer: AB = 10 units
Higher Mathematics
Higher Maths Distance and midpoint
Practise distance and midpoint for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Calculate distances and midpoints accurately and use them inside multi-step geometry problems.
Before you start
- Subtract coordinates in a consistent order before squaring.
- Average x-coordinates and y-coordinates separately.
- Keep exact square roots until a decimal is requested.
Explanation
The distance formula applies Pythagoras to the horizontal and vertical changes between two coordinate points. Squaring removes the signs, but each coordinate difference must still be formed correctly.
The midpoint formula averages corresponding coordinates. It can also be reversed: if one endpoint and the midpoint are known, double each midpoint coordinate and subtract the known endpoint coordinate.
Method and rules
- Distance = √((x₂ − x₁)² + (y₂ − y₁)²)
- Midpoint = (, )
- Missing endpoint coordinate = 2 × midpoint coordinate − known endpoint coordinate
Worked examples
Worked example 2
Find a midpoint
Find the midpoint of C(-5, 4) and D(7, −2).
- Average the x-coordinates: = 1
- Average the y-coordinates: = 1
So: Midpoint = (1, 1)
Worked example 3
Recover a missing endpoint
A(-2, 5) and B have midpoint M(3, −1). Find B.
- For the x-coordinate, = 3, so x = 8.
- For the y-coordinate, = −1, so y = −7.
Answer: B = (8, −7)
Worked example 4
Compare two lengths
P(0, 0), Q(6, 2) and R(2, 6). Show that PQ and PR have equal length
- PQ² = 6² + 2² = 40
- PR² = 2² + 6² = 40
- Equal squared distances give equal positive distances.
So: PQ = PR = √40 = 2√10, so triangle PQR is isosceles
Watch out
- Mixing up midpoint and distance formulae.
- Forgetting the square root after adding the squared coordinate changes.
- Averaging all four coordinates together instead of matching x with x and y with y.
- Rounding a square root too early in a multi-step problem.
Exam reminder
Write the coordinate changes or averages before substituting. If a question asks you to prove lengths are equal, comparing their squared distances is sufficient and avoids unnecessary rounding.
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.