O Level MathematicsE3.5 Equations of parallel and perpendicular lines.

📐 Keep Your Lines in Check! Mastering Parallel & Perpendicular Equations

Edudent Academy
1 Jan 26

Ever wondered how architects ensure walls meet at right angles or why railway tracks never cross? It all boils down to the equations of lines. Understanding parallel and perpendicular lines is **crucial** for O Level success because it links algebra with geometry, helping you solve coordinate-geometry, transformation, and graph-sketching problems quickly in exams.

Slopes That Speak: Parallel vs. Perpendicular

When a line is written in the form y=mx+cy = mx + c, the number mm is its gradient (slope). • **Parallel lines** share the same gradient: if line 1 has m1m_1, then any line parallel to it has m2=m1m_2 = m_1. • **Perpendicular lines** have gradients that multiply to \(-1\): m1×m2=−1m_1 \times m_2 = -1. This means m2=−1m1m_2 = -\dfrac{1}{m_1}. Keep these two rules at your fingertips!

  • Rearrange any linear equation to y=mx+cy = mx + c to read its gradient.
  • For a parallel line, **keep** the same mm.
  • For a perpendicular line, **flip and negate**: m→−1mm \rightarrow -\dfrac{1}{m}.
  • Use the point–slope form y−y1=m(x−x1)y - y_1 = m(x - x_1) when a point (x1,y1)(x_1, y_1) is given.

Worked Example: Perpendicular Through a Point

Problem: Find the equation of the line that passes through the point (4,−2)(4, -2) and is perpendicular to the line y=3x+5y = 3x + 5.

  • Step 1: Identify the gradient of the given line. From y=3x+5y = 3x + 5, we read m1=3m_1 = 3.
  • Step 2: For a perpendicular line, m2=−1m1=−13m_2 = -\dfrac{1}{m_1} = -\dfrac{1}{3}.
  • Step 3: Use point–slope form with (x1,y1)=(4,−2)(x_1, y_1) = (4, -2):
    y−(−2)=−13(x−4).y - (-2) = -\dfrac{1}{3}\bigl(x - 4\bigr).
  • Step 4: Simplify to slope–intercept form:
    y+2=−13x+43y=−13x+43−2y=−13x−23.\begin{aligned} y + 2 &= -\dfrac{1}{3}x + \dfrac{4}{3}\\[6pt] y &= -\dfrac{1}{3}x + \dfrac{4}{3} - 2\\[6pt] y &= -\dfrac{1}{3}x - \dfrac{2}{3}. \end{aligned}

Master these steps and you will glide through coordinate-geometry questions! **Practice** by picking random points and lines, then checking your answers using graphing software or a graphical calculator.