Free Pearson Edexcel International A-level Mathematics · P1

Straight lines and coordinate geometry

What you’ll learn

  • Find line equations from points and gradients.
  • Use parallel and perpendicular conditions to solve geometric problems.

Understand the idea

The gradient measures vertical change divided by horizontal change. Two distinct points determine a non-vertical line, which can be written in point-gradient form. Expand only if the question requests another form; point-gradient form often preserves the geometry more clearly.

Parallel non-vertical lines have equal gradients. For perpendicular lines with finite non-zero gradients, the product of their gradients is −1-1. Horizontal and vertical lines form a separate perpendicular pair. Midpoints and distances help turn geometric conditions into equations.

Key formulae

m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}

y−y1=m(x−x1)y-y_1=m(x-x_1)

m1m2=−1for perpendicular non-vertical linesm_1m_2=-1\quad\text{for perpendicular non-vertical lines}

M=(x1+x22,y1+y22)M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

The perpendicular bisector passes through the midpoint

Conditions. Segment endpoints A = (1,2), B = (7,6), midpoint M = (4,4). The bisector has gradient −3/2. Horizontal and vertical coordinate units use equal screen lengths, preserving perpendicularity.

x / coordinate units; y / coordinate units.

The segment gradient is 2/3, and the bisector gradient is −3/2; their product is −1. The curves meet at (4,4), halfway between A and B. The equal-unit plot also shows their perpendicular directions.

Open the interactive lesson to view plotted curves and coordinate tables.

Worked example

Find the perpendicular bisector of the segment joining (1,2)(1,2) and (7,6)(7,6).

  1. The midpoint is (4,4)(4,4).
  2. The segment has gradient (6−2)/(7−1)=2/3(6-2)/(7-1)=2/3.
  3. A perpendicular line has gradient −3/2-3/2, so use the midpoint in point-gradient form.

Answer: y−4=−32(x−4)y-4=-\frac32(x-4), or 3x+2y=203x+2y=20.

Open the interactive lesson to try original questions and recall. Build a revision plan.