Parallel & Perpendicular Lines – Complete Study Notes

Gradient (Slope) – Core Facts

• Gradient $m$ measures steepness: $m=\frac{\text{rise}}{\text{run}}=\frac{\Delta y}{\Delta x}$.

• Positive $m$: line rises left→right. Negative $m$: line falls left→right.

Parallel Lines

• Key idea: all parallel lines share exactly the same gradient.
  • Examples shown: $y=3x-1$, $y=3x+3$, $y=3x-4$ ⟹ in each case $m=3$.

• To write an equation for a line parallel to a given one:

  1. Copy the gradient $m$ of the given line.

  2. Substitute coordinates of a known point into $y=mx+c$ to find the new intercept $c$.

Perpendicular Lines

• Definition: two lines meet at a right angle (90°).

• Rule connecting their gradients:
  $m<em>1\,m</em>2 = -1 \qquad\text{or equivalently}\qquad m<em>2 = -\frac{1}{m</em>1}$
  • $m<em>2$ is the negative reciprocal of $m</em>1$.
  • Example from the textbook diagram: $m<em>1 = 2\;\;\text{and}\;\;m</em>2=-\tfrac12$ satisfy $2\times(-\tfrac12)=-1$.

• Procedure to find the equation of a perpendicular line:

  1. Compute $m<em>2 = -\tfrac{1}{m</em>1}$.

  2. Insert a known point to solve for $c$ in $y=m_2x+c$.

Quick-Check: Reading Gradients

(From “Building understanding”)

• Gradients of lines parallel to given rules
  a. $y = 4x-6 \Rightarrow m=4$
  b. $y = -7x-1 \Rightarrow m=-7$
  c. $y = -\tfrac13x + 2 \Rightarrow m=-\tfrac13$
  d. $y = 2\tfrac13x - \tfrac11 \Rightarrow m=\tfrac{7}{3}$ (convert mixed number)

• Gradients of lines perpendicular to given rules (use $m<em>2=-1/m</em>1$)
  a. $y=3x-1 \Rightarrow m<em>2=-\tfrac13$ b. $y=-2x+6 \Rightarrow m</em>2=\tfrac12$
  c. $y=\tfrac78x \Rightarrow m<em>2=-\tfrac87$ d. $y=-4x-4 \Rightarrow m</em>2=-\tfrac14$

• Parallel example with intercept:
  • Parallel to $y=5x-2$ through $(0,4)$.
  – $m=5$ (unchanged).
  – Because the point gives the intercept directly, $c=4$.
  – Rule: $y=5x+4$.

• True/False diagnostics
  a. $y=2x$ and $y=2x+3$ → True (same gradient)
  b. $y=3x$ and $y=-3x+2$ → False (product $= -9 \neq -1$)