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7.4 Summary (EMCHX)

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Theorem of Pythagoras:\(AB^2 = AC^2 + BC^2\)
Distance formula:\(AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Gradient:\(m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \quad \text{ or } \quad m_{AB} = \frac{y_1 - y_2}{x_1 - x_2}\)
Mid-point of a line segment:\(M(x;y) = \left( \frac{x_1 + x_2}{2}; \frac{y_1 + y_2}{2} \right)\)
Points on a straight line:\(m_{AB} = m_{AM} = m_{MB}\)
Straight line equationsFormulae
Two-point form:\(\dfrac{y - y_1}{x - x_1} = \dfrac{y_2 - y_1}{x_2 - x_1}\)
Gradient-point form:\(y - y_1 = m (x - x_{1})\)
Gradient-intercept form:\(y = mx + c\)
Horizontal lines:\(y = k\)
Vertical lines\(x = k\)
Parallel lines9daee11f01ada64c3a1ff39e0a0aa32c.png\(m_1 = m_2\)\(\theta_1 = \theta_2\)
Perpendicular linesd82ed7f3faf04f0ef92da6e1fdae45df.png\(m_1 \times m_2 = -1\)\(\theta_{1} = \text{90} ° + \theta_{2}\)
  • Inclination of a straight line: the gradient of a straight line is equal to the tangent of the angle formed between the line and the positive direction of the \(x\)-axis.

    \[m = \tan \theta \qquad \text{ for } \text{0}° \leq \theta < \text{180}°\]

  • Equation of a circle with centre at the origin:

    If \(P(x;y)\) is a point on a circle with centre \(O(0;0)\) and radius \(r\), then the equation of the circle is:

    \[x^{2} + y^{2} = r^{2}\]
  • General equation of a circle with centre at \((a;b)\):

    If \(P(x;y)\) is a point on a circle with centre \(C(a;b)\) and radius \(r\), then the equation of the circle is:

    \[(x - a)^{2} + (y - b)^{2} = r^{2}\]
  • A tangent is a straight line that touches the circumference of a circle at only one point.

  • The radius of a circle is perpendicular to the tangent at the point of contact.