IP EMaths Notes (Upper Sec, Year 3-4): 07) Coordinate Geometry

Coordinate geometry connects algebra with diagrams. These formulas appear in locus questions, perpendicular bisectors, and vector tie-ins.

Core formulas

• Gradient between \((x_1, y_1)\) and \((x_2, y_2)\): \(m = \dfrac{y_2 - y_1}{x_2 - x_1}\) for \(x_1 \neq x_2\).
• Midpoint: \(\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)\).
• Distance: \(d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}\).
• Equation of a line with gradient \(m\) through \((x_0, y_0)\): \(y - y_0 = m(x - x_0)\).

Worked example - Equation of a perpendicular bisector

Points \(A(2, -1)\) and \(B(8, 5)\) form a line segment. Find the equation of the perpendicular bisector of \(AB\).

1. Midpoint of \(AB\): \(M = \left(\dfrac{2 + 8}{2}, \dfrac{-1 + 5}{2}\right) = (5, 2)\).
2. Gradient of \(AB\): \(m = \dfrac{5 - (-1)}{8 - 2} = \dfrac{6}{6} = 1\).
3. Perpendicular slope is \(-1\) (negative reciprocal).
4. Equation using point-slope form: \(y - 2 = -1(x - 5)\).
5. Simplify: \(y - 2 = -x + 5 \Rightarrow x + y - 7 = 0\).

Try this

Determine the coordinates of the point on the line \(3x - 2y = 12\) that is closest to the origin.