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}

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

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

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.

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

m_1 = m_2

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.

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)