Line

Here we take line to be an abbreviation for a straight line.

Euclidean line

A Euclidean line is a Euclidean space of dimension 1, denoted ${\mathbb{E}}^1$. It is a geometric space in which only one real numbers is required to determine the position of each point. It consists of all the points along it. If we have two distinct points $A$ and $B$, then there exists one and only one line that passes through both points.

Slope

The slope is the ratio of $\Delta y$ to $\Delta x$ and is commonly denoted with the letter $m$.

m = \frac{\Delta y}{\Delta x} = \frac{y_{2} - y_{1}}{x_{2}-x_{1}} = \frac{rise}{run}

\begin{document}
\begin{tikzpicture}
    % Top row
    % Slope = 0
    \begin{scope}[shift={(0,3)}]
        \draw[thin] (0,0) -- (1,0);
        \draw[thin] (0,0) -- (0,1);
        \draw[red, thick] (0,0.5) -- (1,0.5);
        \node[below] at (0.5,-0.5) {m = 0};
    \end{scope}
    
    % Slope = 1/2
    \begin{scope}[shift={(3,3)}]
        \draw[thin] (0,0) -- (1,0);
        \draw[thin] (0,0) -- (0,1);
        \draw[red, thick] (0,0) -- (1,0.5);
        \node[below] at (0.5,-0.5) {m = 1/2};
    \end{scope}
    
    % Slope = 1
    \begin{scope}[shift={(6,3)}]
        \draw[thin] (0,0) -- (1,0);
        \draw[thin] (0,0) -- (0,1);
        \draw[red, thick] (0,0) -- (1,1);
        \node[below] at (0.5,-0.5) {m = 1};
    \end{scope}
    
    % Slope = 2
    \begin{scope}[shift={(9,3)}]
        \draw[thin] (0,0) -- (1,0);
        \draw[thin] (0,0) -- (0,1);
        \draw[red, thick] (0,0) -- (0.5,1);
        \node[below] at (0.5,-0.5) {m = 2};
    \end{scope}
    
    % Bottom row
    % Slope = -1/2
    \begin{scope}[shift={(1.5,0)}]
        \draw[thin] (0,0) -- (1,0);
        \draw[thin] (0,0) -- (0,1);
        \draw[red, thick] (0,1) -- (1,0.5);
        \node[below] at (0.5,-0.5) {m = -1/2};
    \end{scope}
    
    % Slope = -1
    \begin{scope}[shift={(4.5,0)}]
        \draw[thin] (0,0) -- (1,0);
        \draw[thin] (0,0) -- (0,1);
        \draw[red, thick] (0,1) -- (1,0);
        \node[below] at (0.5,-0.5) {m = -1};
    \end{scope}
    
    % Slope = -2
    \begin{scope}[shift={(7.5,0)}]
        \draw[thin] (0,0) -- (1,0);
        \draw[thin] (0,0) -- (0,1);
        \draw[red, thick] (0,1) -- (0.5,0);
        \node[below] at (0.5,-0.5) {m = -2};
    \end{scope}
\end{tikzpicture}
\end{document}

The equation for a line is usually of the form:

$$y=mx+c$$

Parallel lines

$L_1$ and $L_2$ are parallel if and only if $m_1=m_2$.

\begin{document} \begin{tikzpicture}[scale=0.8] % First diagram - Parallel Lines \begin{scope}[shift={(0,0)}] % Draw axes \draw[thin] (-1,0) — (4,0) node[right] {x}; \draw[thin] (0,-1) — (0,6) node[above] {y}; % Add tick marks \foreach \y in {0,1,2,3,4,5} \draw (-0.1,\y) — (0.1,\y); \foreach \x in {0,1,2,3} \draw (\x,-0.1) — (\x,0.1); % Draw parallel lines \draw[red, thick] (-0.5,0) — (4,4.5); \draw[blue, thick] (-0.5,-1) — (4,3.5); % Add title \node[right] at (1.5,6) {Parallel Lines}; \end{scope} \end{tikzpicture} \end{document}

Parallel lines have equivalent slopes but differ only where they intercept the y-axis.

Perpendicular lines

$L_1$ and $L_2$ are perpendicular if and only if $m_1=-\frac{1}{m_2}=-m_2^{-1}$. Or equivalently $m_2=-m_1^{-1}$ and $m_1\cdot m_2=-1$.

\begin{document} \begin{tikzpicture}[scale=0.8] \begin{scope} % Draw axes \draw[thin] (-1,0) — (4,0) node[right] {x}; \draw[thin] (0,-1) — (0,6) node[above] {y}; % Add tick marks \foreach \y in {0,1,2,3,4,5} \draw (-0.1,\y) — (0.1,\y); \foreach \x in {0,1,2,3} \draw (\x,-0.1) — (\x,0.1); % Draw perpendicular lines (slope 1 and -1) \draw[red, thick] (0,0) — (4,4); \draw[blue, thick] (0,4) — (4,0); % Add right angle symbol at intersection (2,2), rotated 45 degrees counterclockwise \begin{scope}[shift={(2,2)}, rotate=45] \draw (0,0.4) — (0.4,0.4) — (0.4,0); \end{scope} % Add angle label \node at (2.2,3) {$90^{\circ }$}; % Add title \node[right] at (1.5,6) {Perpendicular Lines}; \end{scope} \end{tikzpicture} \end{document}

Perpendicular lines have a slope that is the negative reciprocal of the partner line.

Ray

If we take a line and some Point $A$ on it, we can consider the line as split into two at the point $A$. Each part is known as a ray. The starting point i.e. $A$ is known as the vertex and the ray consists of all the points along the half-line including $A$.

Line segment

A line segment is part of a straight line that is bounded by two endpoints. It consists of all the points along that line including the endpoints.