Worked example 1
Find a midpoint
Find the midpoint of A(2, −1) and B(8, 5).
- Average the x-coordinates.
- Average the y-coordinates.
Answer: Midpoint = (5, 2)
Back to Higher Mathematics
Higher Mathematics
Distance, midpoint and circles
Use coordinate formulae and interpret (x - a)² + (y - b)² = r².
Before you start
- Square signed differences carefully.
- Average coordinates to find midpoints.
- Recognise the standard circle form.
Explanation
Distance and midpoint formulae describe lengths and halfway points on the coordinate plane. Circle equations use the same geometry: every point on the circle is a fixed distance from the centre.
The standard form (x − a)² + (y − b)² = r² gives centre (a, b) and radius r.
Visual support
Method and rules
- Distance = √((x₂ − x₁)² + (y₂ − y₁)²)
- Midpoint = (x₁ + x₂2, y₁ + y₂2)
- (x − a)² + (y − b)² = r² has centre (a, b) and radius r
Worked examples
Worked example 2
Read a circle equation
State the centre and radius of (x − 3)² + (y + 4)² = 25.
- Compare with (x − a)² + (y − b)² = r²
- The y bracket is y − (-4).
- Take the square root of 25.
So: Centre (3, −4), radius 5.
Watch out
- Mixing up midpoint and distance formulae.
- Forgetting the square root in distance.
- Reading y + 4 as centre y-coordinate 4.
- Using r² as the radius
Exam reminder
Circle questions often hide centre sign changes in brackets. State the centre and radius before doing any tangent or intersection work.