1) The Midpoint Formula
If $M(x,y)$ is the midpoint between the two endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$ then $$ \displaystyle M(x,y)=\left( {\frac{{{{x}_{1}}+{{x}_{2}}}}{2}+\frac{{{{y}_{1}}+{{y}_{2}}}}{2}} \right)$$2) Distance Formula
The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $$ \displaystyle \sqrt{{{{{\left( {{{x}_{2}}-x} \right)}}^{2}}+{{{\left( {{{y}_{2}}-{{y}_{1}}} \right)}}^{2}}}}$$3) The slope of a Line
$$ \displaystyle \text{slope }(m)=\frac{{\text{vertical change}}}{{\text{horizontal change}}}=\frac{{\text{rise}}}{{\text{run}}}$$4) Slope Formula
If the slope of the line passing through any two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is $m$, then $$m=\displaystyle\frac{y_2-y_1}{x_2-x_1}$$5) Positive Slope
A line ascending from left to right has a positive slope.6) Negative Slope
A line descending from left to right has a negaitive slope.7) Zero Slope
The slope of the horizontal line is zero.8) Undefined Slope
The slope of the vertical line is undefined.9) Slope-Intercept Form of a Line
The equation of the form$$y=mx+c$$ is the equation of a straight line with slope $m$ and $y$-intercept $c$, which is called the slope-intercept form.
10) Point-Slope Form of a Line
The equation of a straight line, with slope $m$, and passes through the point $(x_1, y_1)$ is$$y-y_1=m(x-x_1)$$ which is called the point-slope form of a straight line.
11) Horizontal Line
- The $X$-axis and all lines parallel to it are called horizontal lines.
- The equation of horizontal line intersecting the $Y$-axis at $(0,c)$ is $y=c$.
- The equation of the $X$-axis is $y=0$.
12) Vertical Line
- The $Y$-axis and all lines parallel to it are called vertical lines.
- The equation of vertical line intersecting the $X$-axis at $(a,0)$ is $x=a$.
- The equation of the $Y$-axis is $x=0$.
13) Parallel Lines and Perpendicular Lines
- Any two horizontal lines are parallel.
- Any two vertical lines are parallel.
- Vertical and horizontal lines are perpendicular to each other.
14) Some Important Properties
- Two non-vertical lines are parallel if and only if they have the same slope.
- Two non-vertical lines are perpendicular if and only if the product of their slopes is -1 (i.e., one is the negative reciprocal of the other).
- On a same straight line all segments have the same slope.
- Three or more points that lie on the same straight line are said to be collinear.
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