Coordinate Geometry

Within a Cartesian Coordinate System, each point in space has an π‘₯ βˆ’ π‘π‘œπ‘œπ‘Ÿπ‘‘π‘–π‘›π‘Žπ‘‘π‘’ and a 𝑦 βˆ’ π‘π‘œπ‘œπ‘Ÿπ‘‘π‘–π‘›π‘Žπ‘‘π‘’ representing its horizontal position and vertical position. The coordinates are written as an ordered pair, (π‘₯, 𝑦).

1. Distance Formula

The length of a line joining two points can be found based on the Pythagorean Theorem.

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

2. Midpoint Formula

The midpoint of a line joining two points, Β is given by:

M=[\frac{1}{2} (x_2-x_1 ),\frac{1}{2} (y_2-y_1 )]

3. Slope of a Line

The slope of a line is the measure of its gradient at any given point. Β could stand for β€œmodulus of slope.” – the reason is unknown*

Slope,m^*=\frac{Ξ”y}{Ξ”x}

Positive Slope (m>0)

\alpha is an acute angle with the positive sense of the x-axis.
Gradient = tan\theta

Negative Slope (m<0)

\beta is an obtuse angle with the positive sense of the x-axis.
Gradient = tan\theta

In both cases, the gradient of the line passing through A(x_1,y_1 ) and B(x_2,y_2) is given by: \frac{Ξ”y}{Ξ”x}=tan\theta

4. Parallel Lines

If l_1 and l_2 are parallel lines, l_1 and l_2 must be equally inclined to the positive sense of the x-axis. Hence, tan\theta is the slope of l_1 and l_2.

5. Perpendicular Lines

If l_1 and l_2 are perpendicular their slopes must be negative reciprocals of each other:

m_1=-\frac{1}{m_2} OR m_1 m_2=-1

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