Summary
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7.4 Summary (EMCHX)
Theorem of Pythagoras:  \(AB^2 = AC^2 + BC^2\) 
Distance formula:  \(AB = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2}\) 
Gradient:  \(m_{AB} = \frac{y_2  y_1}{x_2  x_1} \quad \text{ or } \quad m_{AB} = \frac{y_1  y_2}{x_1  x_2}\) 
Midpoint 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 equations  Formulae 
Twopoint form:  \(\dfrac{y  y_1}{x  x_1} = \dfrac{y_2  y_1}{x_2  x_1}\) 
Gradientpoint form:  \(y  y_1 = m (x  x_{1})\) 
Gradientintercept form:  \(y = mx + c\) 
Horizontal lines:  \(y = k\) 
Vertical lines  \(x = k\) 
Parallel lines  \(m_1 = m_2\)  \(\theta_1 = \theta_2\)  
Perpendicular lines  \(m_1 \times m_2 = 1\)  \(\theta_{1} = \text{90} ° + \theta_{2}\) 

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.
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