Higher Maths

Straight Line Cheat Sheet

Master the Straight Line unit with this essential guide. Use these formulas and definitions to solve triangle geometry and reasoning problems with confidence.

1

Essential Formulas

Gradient from Two Points
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Equation of a Line
\( y - b = m(x - a) \)
Gradient from Angle
\( m = \tan \theta \)
Midpoint Formula
\( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
2

Gradient Relationships

Parallel Lines
Gradients are equal: \( m_1 = m_2 \)
Perpendicular Lines
Product is -1: \( m_1 \times m_2 = -1 \) or \( m_{\perp} = -\frac{1}{m} \)
Collinearity
Three points are collinear if \( m_{AB} = m_{BC} \) and they share a common point.
3

Triangle Geometry Definitions

Median
Vertex to midpoint of opposite side. Use Midpoint and vertex to find \( m \).
Altitude
Vertex to opposite side at \( 90^\circ \). Use perpendicular gradient of opposite side.
Perpendicular Bisector
Midpoint of a side at \( 90^\circ \). Use Midpoint and perpendicular gradient.
4

Exam Technique & Notation

Perpendicular Check
To prove lines are perpendicular, you must show that \( m_1 \times m_2 = -1 \).
Undefined Gradient
Vertical lines (\( x_1 = x_2 \)) have an undefined gradient; form is \( x = k \).
Intersection
Find where lines meet by solving their equations simultaneously.
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SQA Exam Tips

  • Exact Values: Always leave gradients as simplified fractions or surds; do not convert to decimals unless the question involves a calculator.
  • Show Working: Always state the midpoint and gradient explicitly before substituting into \( y - b = m(x - a) \).
  • Negative Gradients: Be careful with the tangent of obtuse angles; remember that \( m = \tan \theta \) is negative when \( 90^\circ < \theta < 180^\circ \).