6.1 Midpoints and Perpendicular Bisectors

- The **midpoint** M of a line segment connecting points A(x₁, y₁) and B(x₂, y₂) is given by:

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

- It gives the exact halfway point between two coordinates.

- A **perpendicular bisector** of a line segment:

- Passes through the midpoint of the segment

- Is **perpendicular** to the original line

- To find the equation of the perpendicular bisector:

1. Find the **midpoint** of the segment

2. Find the **negative reciprocal** of the original gradient (m → -1/m)

3. Use point-slope form: y - y₁ = m(x - x₁)

- A = (2, 4), B = (6, 8)

1. Midpoint = ((2+6)/2, (4+8)/2) = (4, 6)

2. Gradient of AB = (8 - 4)/(6 - 2) = 1 → Perpendicular gradient = -1

3. Equation: y - 6 = -1(x - 4) → y = -x + 10

- Perpendicular bisectors are used in:

- Finding centres of circles

- Triangle circumcentres

- Construction and symmetry problems