Coordinate Geometry

Distance Between Two Points

Using a right-angled triangle and following Pythagoras’ theorem the distance between two points (hypotenuse) can be found.

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Midpoint of a Line

(x_3, y_3) = (\frac{x_1 + x_2}{2}), (\frac{y_1 + y_2}{2})

Gradient of a Straight Line

m = \frac{y_2-y_1}{x_2-x_1}

Equation of a Straight Line

y = mx + c

Equation when the gradient and one point are known:

y-y_1 = m(x-x_1)

Equation when two points are known:

\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}

Other form of straight line equation:

ax + by +c = 0

Any equation which can be arranged into the form ax + by + c = 0 or y = mx + c is a straight line.

Intercepts Equation of a Straight Line

Rearrange the straight line equation to equal the below. To find the y-intercept x=0. To find the x-intercept y=0.

\frac{x}{a} + \frac{y}{b} = 1

Parallel Lines

m_1 = m_2

Perpendicular Lines

The product of two perpendicular lines is -1.

m_1 = \frac{-1}{m_2}

This cannot be applied for horizontal (gradient = 0) or vertical (gradient = undefined) lines as m₁ \x m₂ has no value.

Altitudes

In a triangle ABC, the altitude through A is the line perpendicular to BC which goes through A.

Distance of a Point From a Line

The perpendicular distance of a point from a line is always the shortest and can be calculated with:

d = \left\|\frac{aA + bB + c}{\sqrt{a^2+b^2}}\right\|

Graphical Methods

Definitions

Relation — set of ordered pairs of numbers (x,y)

Function — a relation with no repetitions of x-values in the ordered pairs

Domain — numbers which can be entered into a function

Range — numbers which result from the function rule being applied

Asymptote — a line to which a graph gets ever closer but never actually touches

Periodic Functions

A function is periodic if, for some number p, f(x) = f(x+p). The minimum positive value of such numbers p is the period of f. A periodic graph will repeat itself over every interval of width p.

Fractions

\frac{a}{b} / \frac{c}{d} = \frac{a}{b} \x \frac{d}{c}

\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

\frac{a}{b} = ab⁻¹

Determine if a Quadratic has Real or Complex Roots

D=b^2−4ac

For a quadratic

ax^2+bx+c=0

If D>0: two real solutions

If D=0: one real solution (repeated root)

If D<0: no real solutions → complex roots

Midpoint theorem — https://www.youtube.com/watch?v=gutMr88XPcE

https://en.wikipedia.org/wiki/Midpoint_theorem_(triangle)