Midpoint of a Line Segment

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: \(\big(\tfrac{x_1+x_2}{2}, \tfrac{y_1+y_2}{2}\big)\).