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Higher Maths · Expressions and Functions

The Straight Line | Higher Maths

The Straight Line is the foundation topic in Higher Maths. It builds on National 5 work and underpins almost every other topic — from differentiation to vectors. Mastering gradient, perpendicular bisectors and altitudes is essential before you tackle the calculus sections of Paper 1 and Paper 2.

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SQA Higher MathsSpecification: Expressions and FunctionsUnit 1 (legacy)

Core Straight Line formulas

Gradient between two points: m = (y2 − y1) / (x2 − x1)
Equation of a line through (a, b) with gradient m: y − b = m(x − a)
Gradient and angle: m = tan θ
Parallel lines: m1 = m2
Perpendicular lines: m1 × m2 = −1
Midpoint of (x1, y1) and (x2, y2): M = ((x1+x2)/2, (y1+y2)/2)

Worked example

Worked example — Perpendicular bisector

Problem: Find the equation of the perpendicular bisector of the line joining A(−1, 4) and B(5, −2).

  1. Find the midpoint M of AB.
    M = ((−1+5)/2, (4+(−2))/2) = (2, 1)
  2. Find the gradient of AB.
    mAB = (−2 − 4) / (5 − (−1)) = −6/6 = −1
  3. Gradient of the perpendicular bisector is the negative reciprocal.
    mperp = 1 (since −1 × 1 = −1)
  4. Use point-gradient form with M(2, 1) and m = 1.
    y − 1 = 1(x − 2) ⇒ y = x − 1

Practice questions

Try these SQA-style questions. Tap "Show answer" to check your working.

Practice questions

  1. Find the gradient of the line joining (3, −2) and (−1, 6).
    Show answer
    m = −2 (working: (6−(−2))/(−1−3) = 8/−4)
  2. Determine whether the lines y = 2x + 5 and 2y + x = 7 are perpendicular.
    Show answer
    Yes. m1 = 2, m2 = −1/2, product = −1.
  3. Find the equation of the line through (4, −1) parallel to y = −3x + 2.
    Show answer
    y = −3x + 11
  4. Triangle PQR has P(1, 2), Q(7, 4), R(3, 8). Find the equation of the altitude from R.
    Show answer
    Gradient PQ = 1/3, so altitude gradient = −3. Equation: y − 8 = −3(x − 3) ⇒ y = −3x + 17
  5. A line makes an angle of 60° with the positive x-axis. State its gradient.
    Show answer
    m = tan 60° = √3

Common mistakes

Common mistakes & how to avoid them

  • Forgetting that perpendicular gradients multiply to −1 (not add).
  • Mixing up the order of subtraction in the gradient formula — be consistent: top and bottom must use the same point as the "first" point.
  • Using the gradient of the original line instead of the perpendicular gradient when finding altitudes or perpendicular bisectors.

Frequently asked questions

Is the straight line tested on Paper 1 or Paper 2?
Both papers can include straight-line questions. Paper 1 (non-calculator) typically tests algebraic gradient and equation work. Paper 2 (calculator allowed) often combines the straight line with circles or trigonometry.
Do I need to memorise the gradient formula?
Yes. The gradient formula is not on the SQA formula sheet. You must know m = (y2 − y1) / (x2 − x1) by heart.
What is the difference between a median, altitude and perpendicular bisector?
A median goes from a vertex to the midpoint of the opposite side. An altitude is perpendicular to the opposite side from a vertex. A perpendicular bisector is perpendicular to a side and passes through its midpoint.

Related Higher Maths topics

These topics often appear together in SQA exam questions.

Functions & GraphsDifferentiationThe CircleVectors

← All Higher Maths topics

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