Midpoint of a Line Segment

\( \left( \tfrac{x_1+x_2}{2},\; \tfrac{y_1+y_2}{2} \right) \)
Coordinate Geometry GCSE
Question 1 of 20
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Find the midpoint of (–2,4) and (6,–4).

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
Average the x and y coordinates.

Explanation

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Statement

The midpoint of a line segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\) is given by:

\[ \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \]

Why it’s true

  • The midpoint lies exactly halfway between the two endpoints in both the x and y directions.
  • Averaging the x-coordinates gives the horizontal middle.
  • Averaging the y-coordinates gives the vertical middle.

Recipe (how to use it)

  1. Take the two x-values, add them, divide by 2.
  2. Take the two y-values, add them, divide by 2.
  3. The result is the midpoint coordinate.

Spotting it

This is used in coordinate geometry when asked to find the midpoint of a line, diagonal of a shape, or centre of a segment.

Common pairings

  • Geometry problems involving triangles and quadrilaterals.
  • Coordinate geometry questions on graphs.
  • Finding centres of symmetry.

Mini examples

  1. Endpoints (2,4) and (6,8). Midpoint = ((2+6)/2, (4+8)/2) = (4,6).
  2. Endpoints (–3,5) and (1,–1). Midpoint = ((–3+1)/2, (5+–1)/2) = (–1,2).

Pitfalls

  • Mixing up formula: Don’t subtract, always add and halve.
  • Arithmetic slips: Be careful with negatives and fractions.
  • Wrong order: Keep x’s together and y’s together.

Exam strategy

  • Write coordinates clearly.
  • Do x’s and y’s separately.
  • Check: midpoint must lie between the two points on the line.

Summary

The midpoint of \((x_1,y_1)\) and \((x_2,y_2)\) is the average of their coordinates: \(\big(\tfrac{x_1+x_2}{2}, \tfrac{y_1+y_2}{2}\big)\).

Worked examples

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  1. Find the midpoint of (2,4) and (6,8).
    1. \( Midpoint= ((2+6)/2, (4+8)/2). \)
    2. \( (8/2, 12/2)=(4,6). \)
    Answer: (4,6)
  2. Find the midpoint of (–3,5) and (1,–1).
    1. \( x=(–3+1)/2=–1. \)
    2. \( y=(5+–1)/2=2. \)
    3. \( Midpoint=(–1,2). \)
    Answer: (–1,2)
  3. Find the midpoint of (0,0) and (10,0).
    1. \( x=(0+10)/2=5. \)
    2. \( y=(0+0)/2=0. \)
    3. \( Midpoint=(5,0). \)
    Answer: (5,0)
  4. Find the midpoint of (–4,–6) and (2,2).
    1. \( x=(–4+2)/2=–1. \)
    2. \( y=(–6+2)/2=–2. \)
    3. \( Midpoint=(–1,–2). \)
    Answer: (–1,–2)
  5. Find the midpoint of (3,7) and (11,3).
    1. \( x=(3+11)/2=7. \)
    2. \( y=(7+3)/2=5. \)
    3. \( Midpoint=(7,5). \)
    Answer: (7,5)
  6. Find the midpoint of (–2,10) and (4,–2).
    1. \( x=(–2+4)/2=1. \)
    2. \( y=(10+–2)/2=4. \)
    3. \( Midpoint=(1,4). \)
    Answer: (1,4)
  7. Find the midpoint of (1,1) and (7,9).
    1. \( x=(1+7)/2=4. \)
    2. \( y=(1+9)/2=5. \)
    3. \( Midpoint=(4,5). \)
    Answer: (4,5)
  8. Find the midpoint of (–8,4) and (–2,12).
    1. \( x=(–8+–2)/2=–5. \)
    2. \( y=(4+12)/2=8. \)
    3. \( Midpoint=(–5,8). \)
    Answer: (–5,8)
  9. Find the midpoint of (0,–3) and (6,3).
    1. \( x=(0+6)/2=3. \)
    2. \( y=(–3+3)/2=0. \)
    3. \( Midpoint=(3,0). \)
    Answer: (3,0)
  10. Find the midpoint of (–5,–5) and (5,5).
    1. \( x=(–5+5)/2=0. \)
    2. \( y=(–5+5)/2=0. \)
    3. \( Midpoint=(0,0). \)
    Answer: (0,0)