O-Level Coordinate geometry in two dimensions: worked solution

Question

With h = 2, giving A(0, 2) and B(4, -2), obtain the equation of the line that perpendicularly bisects AB.

Method: a perpendicular bisector does two things at once. It passes through the midpoint of AB, and it is perpendicular to AB. So find the midpoint, find the gradient of AB, take the negative reciprocal for the perpendicular gradient, then use y − y₁ = m(x − x₁). Step 1 Midpoint of A(0, 2) and B(4, −2). Average each coordinate. x-coordinate: (0 + 4)/2 = 4/2 = 2 y-coordinate: (2 + (−2))/2 = 0/2 = 0 Midpoint M(2, 0) Step 2 Gradient of AB, using m = (y₂ − y₁)/(x₂ − x₁). m(AB) = (−2 − 2)/(4 − 0) = −4/4 = −1 Step 3 Gradient of the perpendicular. For perpendicular lines m₁ × m₂ = −1, so m₂ = −1/m₁ = −1/(−1) = 1 Step 4 Equation of the line through M(2, 0) with gradient 1, using y − y₁ = m(x − x₁). y − 0 = 1(x − 2) y = x − 2 Check: M(2, 0) satisfies it, since 2 − 2 = 0 ✓, and its gradient 1 is the negative reciprocal of −1 ✓. So the perpendicular bisector is y = x − 2.