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Cambridge IGCSE Mathematics — 0580 Extended

Topic 3.4: Coordinate geometry — Length & Midpoint

Length of a line segment

The distance between \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is the hypotenuse of a right-angled triangle with sides \(\lvert x_2 - x_1\rvert\) and \(\lvert y_2 - y_1\rvert\):

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Method

  1. Find \(\Delta x = x_2 - x_1\) and \(\Delta y = y_2 - y_1\).
  2. Square both, add, then take the square root.
  3. If the result is not a perfect square, leave the answer in surd form (especially on Paper 2).
Right triangle on a grid showing length 5 between A(1,1) and B(4,5)
Classic \(3\)-\(4\)-\(5\): \(d = \sqrt{3^2 + 4^2} = 5\).

Find the length of \(AB\) where \(A(2, 1)\) and \(B(5, 3)\).

Segment from A(2,1) to B(5,3) with length square root of 13
\(d = \sqrt{3^2 + 2^2} = \sqrt{13}\).

Paper 2 (non-calculator)

If the number under the root is not a perfect square, leave the answer as a surd (e.g. \(\sqrt{13}\)). Do not invent a decimal approximation unless the question asks for one.

Midpoint of a line segment

The midpoint \(M\) is halfway between \(A\) and \(B\) — average the coordinates:

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

Method

  1. Add the \(x\)-coordinates and divide by \(2\).
  2. Add the \(y\)-coordinates and divide by \(2\).
  3. Write the answer as an ordered pair \((x, y)\).

Find the midpoint of \(A(1, 2)\) and \(B(5, 8)\).

Segment A(1,2) to B(5,8) with midpoint M(3,5) marked
\(M = \left(\dfrac{1+5}{2},\; \dfrac{2+8}{2}\right) = (3, 5)\).
Formula board showing length and midpoint formulas
Keep both formulas ready for Paper 2 and Paper 4.

Try this

\(C(-3, 4)\) and \(D(5, -2)\). Find the length of \(CD\) and the midpoint of \(CD\).

Show answer
Answer

  1. Differences

    \[ \Delta x = 5 - (-3) = 8,\quad \Delta y = -2 - 4 = -6 \]

  2. Length

    \[ d = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

  3. Midpoint

    \[ M = \left(\frac{-3+5}{2},\; \frac{4+(-2)}{2}\right) = (1, 1) \]

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