O Level MathematicsE3.4 Length and midpoint of a line segment (use formulas).

📏 Unlocking Line Segments: Length & Midpoint Made Easy!

Edudent Academy
30 Dec 25

Ever wondered how far apart two points on a coordinate grid really are, or exactly where the centre of that line segment lies? Understanding the **length** and **midpoint** of a line segment is vital for tackling coordinate geometry questions in your O Level exams—and it’s surprisingly straightforward once you know the formulas!

The Key Formulas at a Glance

Given two points A(x1,y1)A(x_1, y_1) and B(x2,y2)B(x_2, y_2), the **distance (length) formula** is
AB=(x2−x1)2+(y2−y1)2AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
, and the **midpoint** is
M(x1+x22,  y1+y22).M\Bigl(\dfrac{x_1 + x_2}{2},\; \dfrac{y_1 + y_2}{2}\Bigr).
These formulas come directly from Pythagoras’ theorem and simple averaging—so if you remember triangles and mean averages, you’re already halfway there!

  • The length formula finds the straight-line distance between two points.
  • The midpoint formula locates the exact centre point of a line segment.

Worked Example: Bridging Two Points

Problem: Find the length of the line segment joining A(2,−1)A(2, -1) and B(8,5)B(8, 5), and determine its midpoint.

  • Step 1: Calculate the length using
    AB=(8−2)2+(5−(−1))2=62+62=72=62  units.AB = \sqrt{(8 - 2)^2 + (5 - (-1))^2} = \sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2}\; \text{units}.
  • Step 2: Find the midpoint using
    M(2+82,  −1+52)=M(5,2).M\Bigl(\dfrac{2 + 8}{2},\; \dfrac{-1 + 5}{2}\Bigr) = M(5, 2).

Practice makes perfect—so grab a set of random points, apply these formulas, and soon you’ll be calculating lengths and midpoints **with confidence**. Keep refining your skills, and you’ll ace every coordinate geometry question that comes your way!