7.4 Summary | Analytical geometry

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:	$m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \quad \text{ or } \quad m_{AB} = \frac{y_1 - y_2}{x_1 - x_2}$
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:	$m_{AB} = m_{AM} = m_{MB}$

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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