Parallel & Perpendicular Lines

Two gradient facts carry this subtopic:

• Parallel: $m_1 = m_2$
• Perpendicular: $m_1 m_2 = -1$, gradients are negative reciprocals ($m \to -\tfrac{1}{m}$)

Flip and negate: $\tfrac{2}{3} \to -\tfrac{3}{2}$, $-4 \to \tfrac{1}{4}$, $1 \to -1$. Both facts are assumed knowledge in half the coordinate-geometry questions on the paper.

Building lines from the conditions

Find the line through $(4, 1)$ perpendicular to $2x + 3y = 6$. Rearrange to read the gradient: $y = -\tfrac{2}{3}x + 2 \to m = -\tfrac{2}{3}$ Perpendicular gradient: $m' = \tfrac{3}{2}$ (state the negative-reciprocal step, it’s the M mark) Line: $y - 1 = \tfrac{3}{2}(x - 4) \to$ $2y = 3x - 10$ (the build routine)

The only reliable way to read a gradient from $ax + by = c$ form is to rearrange, eyeballing coefficients produces sign errors under pressure.

Proving geometry with gradients

The conditions turn shape claims into gradient arithmetic:

• “Show $AB$ is parallel to $CD$”: compute both gradients, show equal, state the conclusion
• “Show angle $ABC$ is $90°$”: gradients of $BA$ and $BC$, product $= -1$, conclusion in words
• “Show $ABCD$ is a trapezium/parallelogram/rectangle”: the right pairs of sides parallel (and perpendicular, for the rectangle), list which pairs you’re testing before computing, so the logic reads as an argument rather than a heap of fractions

Each gradient is an M/A mark; the stated conclusion (“since $m_{AB} \times m_{BC} = -1$, angle $ABC = 90°$”) is its own mark and the most commonly omitted one.

Where the conditions hide

Perpendicularity is the syllabus’s favourite smuggled ingredient: normals to curves (normal $\perp$ tangent), tangents to circles (tangent $\perp$ radius), and perpendicular bisectors. In each, the visible step “gradient of perpendicular $= -\tfrac{1}{m}$” earns method credit, write it every time, even when it feels obvious.

Common mistakes

• Reciprocal without the sign flip (or the flip without the reciprocal)
• Gradients read from unrearranged $ax + by = c$
• Horizontal/vertical pairs fumbled (perpendicular to $y = 3$ is $x = k$, not anything with $m_1 m_2 = -1$, the rule fails when one gradient is $0$ or undefined; say it in words instead)
• Geometric conclusions computed but never stated
• The wrong vertex’s angle tested in “show the right angle is at $B$”

Full topic context: Straight-Line Graphs notes.