Sheet № 146 · Foundation + Higher · AQA · Edexcel · OCR
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 a
§Key definitions
Question:
Find the midpoint of (2, 6) and (8, 10).
Answer:
B = (8, −1)
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).
§Formulas to memorise
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Distance = √((x₂ − x₁)² + (y₂ − y₁)²)
Add the two x-coordinates — and divide by 2.
Add the two y-coordinates — and divide by 2.
Write the midpoint — as a coordinate pair.
Subtract the x-coordinates — to find the horizontal difference.
Subtract the y-coordinates — to find the vertical difference.
Square each difference — , add them together, and take the square root.
Worked example
Find the midpoint of (2, 6) and (8, 10).
Working:
⚠ 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).